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Mechanochemical Processes in

Energetic Materials: A

Computational and Experimental

Investigation

Adam Alexander Leon Michalchuk

Submitted for the Degree of Doctor of Philosophy

University of Edinburgh

August 2018

ii

Abstract

Energetic materials (explosives, propellants and pyrotechnics; EMs)

encompass a broad range of materials. These materials are used across a

wide spectrum of applications, including civil and defence. For example, HMX,

RDX and TNT are well known EMs with defence applications. Silver fulminate

is instead used in house-hold Christmas crackers and ammonium nitrate is

used for numerous industrial applications. Common to all EMs is their

propensity to rapidly release energy upon external perturbation. The amount

and type of energy that is required to initiate an EM can vary across orders of

magnitude. Some materials (e.g. triaminotriperoxide, TATP) initiate with < 1 J

of impact energy, while others (e.g. triaminotrinitrobenzene, TATB) cannot be

initiated without > 100 J of impact energy. Understanding which materials can

be handled safely is therefore of critical importance for maintaining the safe

use of EMs across all sectors.

Current trends in EM research include a drive to develop new materials with

decreased sensitivities. While it is relatively straightforward to selectively

modify some properties (e.g. environmental impact), very little is understood

about what constitutes a sensitive material. At present, a new EM must be

synthesised and its sensitivity tested. However, with no a priori knowledge of

the potential sensitivity of a novel EM, synthesis is accompanied by substantial

hazard, as well as time and financial costs. It is therefore pressing to develop

a fundamental understanding of what dictates a sensitive material, and hence

develop a mechanism to predict these properties. A particularly promising

model to explore impact sensitivity of EMs is based on vibrational up-pumping,

i.e. the up-conversion of vibrational energy. This thesis explores the

application of this model to a set of azide, organic molecular and polymorphic

materials.

Azide-based EMs share the common N3− explosophore. The electronic

structure of this anion was followed as a function of its normal modes of

iii

vibration. It was found that excitation of the bending mode is sufficient to induce

athermal electronic excitation of the molecule, and spontaneous

decomposition. This is valid both in the gas and solid states. It is therefore

suggested that this vibrational mode is largely responsible for decomposition

of the azide materials. Based on calculations of the complete phonon

dispersion curves, the various pathways to vibrational energy up-pumping

were explored, namely via overtone and combination pathways. In particular,

the relative rates of up-pumping into the N3− bending mode were investigated.

Remarkable agreement is found between these up-pumping rates and the

relative ordering of the impact sensitivity of these azides.

The calculated vibrational structures of organic molecular EMs were first

compared with experimental inelastic neutron scattering spectra and found to

provide accurate representation of the low temperature vibrational structure of

these complex crystals. The decomposition pathways for organic EMs are not

known and hence no target frequency could be unambiguously identified.

Instead, the up-pumping model was developed for these materials by

investigating the total rate of energy conversion into the internal vibrational

manifold. A number of qualitative trends were identified, which may provide a

mechanism for the rapid classification of EMs from limited vibrational

information. The overtone pathways were found to offer a good agreement with

experimental impact sensitivities of these compounds. However, the increased

complexity of the vibrational structure of the organic EMs as compared to the

azides required a more thorough treatment of the up-pumping mechanism to

correctly reflect experimental sensitivities. The effects of temperature on up-

pumping were also explored.

The sensitivity of organic EMs is known to differ across polymorphic forms.

Most notable are the HMX polymorphs. The calculated vibrational structure of

two HMX polymorphs was confirmed by inelastic neutron scattering

spectroscopy. The up-pumping model developed for molecular organic EMs

was therefore extended to a comparison of these two HMX polymorphs. The

polymorphic forms of FOX-7 were also investigated under the premise of the

iv

up-pumping model. Upon heating, FOX-7 undergoes two polymorphic

transformations, which increases the layering of the materials. It therefore

offered an opportunity to explore the widely-held hypothesis that layered

materials are less sensitive than non-layered materials. The metastable γ-form

was successfully recovered, and its experimental impact sensitivity

investigated by BAM drop-hammer method. However, upon impact, the γ-

polymorph appeared to convert to the α-form and initiate at the same input

energy. Hence a considerable deficiency of experimental methods is identified

when studying polymorphic materials. FOX-7 was therefore explored within the

framework of the up-pumping model. The inelastic neutron scattering spectrum

was collected for γ-FOX-7, which confirmed the calculated vibrational structure.

It was shown that within the up-pumping model, the layered γ-polymorph is

predicted to be less sensitive than the α-form, and results from a decrease in

the maximum phonon-bath frequency. Hence a new mechanism is proposed

to describe the insensitivity of layered compounds.

The work presented in this thesis explores the applications of vibrational up-

pumping to rationalise and predict the relative impact sensitivities of a range

of EMs. Despite the approximations employed in construction of the model, it

leads to excellent correlation with experimental results in all cases. This work

therefore opens the door to a new fully ab initio approach to designing new

EMs based solely on knowledge of the solid-state structure.

v

Lay Summary

The term energetic material (EM) describe a range of compounds, which can

be broadly classified as explosives, propellants and pyrotechnics. While it is

most common to think of these materials as having military-based applications,

their use is far more widespread. Christmas crackers contain a primary

explosive known as silver fulminate, fireworks contain both pyrotechnics and

propellants, and many automobile airbags are based on a range of EMs. Many

industries are also dependent on the use of EMs, including the mining and oil

industries. Common to all EMs is their propensity to explode when physically

struck, heated or subjected to an electric shock. The amount of energy that is

required depends strongly on the material. This is a critical physical parameter

for ensuring the safe use of EMs across any sector.

With growing pressures to produce new EMs (e.g. for environmental purposes),

effort is being placed on ensuring new EMs exhibit low sensitivities, i.e. are

safer. It is relatively straightforward to design new materials with better

environmental compatibility. For example, by removing lead or other heavy

metals. However, there is no definitive understanding for what dictates the

sensitivity of EMs. Any new EM must be fully synthesised in relatively large

quantities, and tested. With no preceding knowledge of the physical properties

of a new EM, this can be very dangerous. There is therefore considerable

interest in developing computational methods to rationalise and predict the

sensitivity of EMs.

When a material is struck by a physical blow, it introduces a large amount of

energy into the material. Analogous to water, an impact leads to formation of

waves, which dissipate through the medium. In a solid material, this translates

into the vibrations of the molecules. This energy subsequently dissipates

through the material and activates a chemical bond whose rupture leads to

initiation and explosion. This thesis explores this mechanism as an underlying

model for the rationalisation and prediction of impact (mechanical) sensitivity

vi

of a range of EMs, include N3− -based materials and organic molecular

materials.

The bond which ruptures in N3− -based materials is an N-N bond. A series of

quantum mechanical calculations demonstrated that if the linear N3− molecule

is bent, dissociation becomes favourable. Therefore, impact sensitivity of these

materials was investigated by considering the rate of vibrational energy

dissipation into the bending vibrational mode. This led to excellent correlation

with experimental result. A similar approach was taken to investigate organic

molecular EMs. However, given the complex molecular structures of these

compounds, it was not possible to identify a single vibration that is responsible

for dissociation. Instead, predictions were based on the total rate at which

energy dissipates into the molecule. Again, excellent agreement is obtained

with experimental results.

Many organic molecules can crystallise into different crystalline forms.

Composed of the same molecule, these solid forms have different

intermolecular interactions, which can lead to considerably different physical

properties. Different polymorphic forms of EMs are also known to exhibit

different sensitivities, most notably HMX. The vibrational energy transfer model

was therefore applied to HMX polymorphs, and shown to offer successful

differentiation of two polymorphic forms. The polymorphs of FOX-7 were

therefore also investigated. It is typically accepted that crystals that contain

layers of molecules (e.g. TATB) are very insensitive. On heating, the

polymorphic forms of FOX-7 exhibit increased layering, and the high

temperature form is therefore expected to be less sensitive. However, when

the high temperature form was tested, it was found to convert to the ambient

temperature form upon impact. Experiment is therefore not able to measure

the sensitivities of these forms, and may in fact produce erroneous results for

other polymorphic materials. The vibrational energy conversion model,

however, does predict that the layered (high temperature) compound should

be less sensitive than the ambient temperature (non-layered) form, and offers

a new mechanism for rationalising why layered materials are insensitive.

vii

The work in this thesis develops a new approach for understanding the

mechanical impact sensitivity of a range of EMs. Based purely on

computational methods, this work demonstrates that it may in fact be possible

to predict the sensitivity properties of new EMs without the need for potentially

dangerous synthetic procedures.

viii

Declaration

I declare that this thesis was written by myself and that the work detailed in this thesis

is my own, or I have contributed substantially to such work, except where specific

reference is made to the work of another.

Adam A. L. Michalchuk

ix

“I am just a child who has never grown up. I still keep

asking these ‘how’ and ‘why’ questions… Occasionally, I

find an answer’’

-Prof. Stephen Hawking

x

Acknowledgements

There are very many people I must thank, without whom this would not have

been possible.

First and foremost, to my supervisors Prof. Colin Pulham and Dr Carole

Morrison. I could not have asked for a better team of mentors to guide me

through my PhD. To Colin, who accepted my transfer into the School of

Chemistry many years ago, sent me to Siberia time and again, gave me the

opportunity to pursue my PhD and introduced me to the world of energetic

materials. To Carole, for adopting me as her student when my project moved

towards computational chemistry and for teaching me so much about the field.

To you both for your enthusiasm and encouragement: thank you.

My sincerest thanks to all of the members of the Pulham group, past and

present, for making the office feel like a family. To Dan, Emily, Hayleigh, Nisa,

Oleg, Rowan, Stuart, Sumit, Xiaojiao– you have all made the last four years

amazing, from laughs around the office to pub quizzes, games nights and a

get-away to Arran. Special thanks to Karl, for our many cross-country (and

indeed cross-continental!) synchrotron road trips, and for not getting too

distracted by trains while driving. To Rowan, Hayleigh and Xiaojiao for

welcoming me to the group four years ago. And thanks to Nilgun and Stuart

for hilarious trips to the Cavendish Laboratory. My thanks also to Prof. Adam

Cumming for his guidance. I am also very grateful for the friendship of the

Boldyreva group (Novosibirsk, Russia) for their warm hospitality during my

many stays in Novosibirsk. A special thanks to Prof Elena Boldyreva for her

guidance and mentorship, to Academician Prof. Vladimir Boldyrev for many

stimulating discussions and Prof Andrei Arzhannikov for his hospitality and

friendship.

xi

I must also thank the EPSRC Centre for Continuous Manufacturing and

Crystallisation (CMAC) and an Edinburgh Global Research Scholarship for

funding my PhD studies. Thank you to my fellow CMAC cohort (Alex, Alice,

Antonia, Arabella, Bilal, Bruce, Carlotta, Lauren, Meifen, Ravi, Sara, Vaclav)

and the rest of CMAC for making these years so enjoyable.

Further thanks to Dr. Svemir Rudić (STFC ISIS) for allowing me to put

explosives on the TOSCA beamline, for his enthusiasm and for teaching me

much about inelastic neutron scattering. Thanks to Dr Steven Hunter

(University of Edinburgh) for helping me get started in computational chemistry.

Additional thanks to Dr David Williamson (Cavendish Laboratory, University of

Cambridge) for access to the BAM fall hammer.

Thank you to Dom for your friendship over the past four years, and for always

being ready for a good laugh. To my fellow Canadian, Paul, thank you for your

friendship and of course for the Canadian care packages. Darren, my sincerest

thanks for putting up with my constant questions about Molpro, and for helping

me so much with its use.

And to everyone else who has made my nine years in Edinburgh so

unforgettable, my deepest thanks.

Finally, to my family back in Canada – this would not have been possible

without you. Thank you.

- Adam

xii

Abbreviations

ABT 1,1’-Azobistetrazole

BOA Born-Oppenheimer Approximation

CASTEP Cambridge Serial Total Energy Package

CI Configurational Interaction

CL-20 Hexanitrohexaazaisowurtzitane (HNIW)

DFPT Density Functional Perturbation Theory

DFT Density Functional Theory

D2 Grimme’s D2 dispersion correction

EM Energetic Material

FC Frank-Condon

FOX-7 1,1-diaminio-2,2-dinitroethene (DADNE)

GGA Generalised Gradient Approximation

G06 Grimme’s D2 dispersion correction

HBT 5,5’-Hydrazinebistetrazole

HF Hartree-Fock

HK Hohenberg-Kohn

HMX 1,3,5,7-Tetranitro-1,3,5,7-tetrazocane (Octagen)

HNB Hexanitrobenzene

INS Inelastic Neutron Scattering Spectroscopy

KS Kohn-Sham

LBS Localised Basis Set

LDA Local Density Approximation

MP Monkhorst-Pack

NTO Nitrotriazolone

PETN Pentaerythritoltetranitrate

PW Plane Wave

pwDFT Plane Wave Density Functional Theory

TATB Triamino-trinitrobenzene

TATP Triacetone-triperoxide

TNT 2,4,6-trinitrotoluene

TS Tkachenko-Scheffler dispersion correction

xiii

Contents

INTRODUCTION ................................................................................................................................ 1

1.1 ENERGETIC MATERIALS ....................................................................................................................... 1

1.1.1 Energetic Materials: A Brief History ...................................................................................... 1

1.1.2 Insensitive Munitions ............................................................................................................ 3

1.1.3 Energetic Materials: Definitions and Classifications ............................................................. 4

1.2 INITIATION OF ENERGETIC MATERIALS ................................................................................................... 8

1.2.1 Hot Spot Models .................................................................................................................... 8

1.2.2 Vibrational Up-Pumping ...................................................................................................... 11

1.3 PREDICTION AND RATIONALISATION OF ENERGETIC MATERIAL SENSITIVITY ................................................. 15

1.3.1 Isolated Molecule Methods ................................................................................................ 15 1.3.1.1 Empirical Fitting of Molecular Descriptors ................................................................................... 16 1.3.1.2 Oxygen Balance ............................................................................................................................ 17 1.3.1.3 NMR Chemical Shift ...................................................................................................................... 19 1.3.1.4 Bond Energies and Dissociation .................................................................................................... 20

1.3.2 Solid State Methods ............................................................................................................ 22 1.3.2.1 Crystal Packing and Non-Covalent Interactions ............................................................................ 23 1.3.2.2 Electronic Band Gap Criterion and Band Gap Dynamics ............................................................... 26

1.3.3 Kinetic Models ..................................................................................................................... 27

1.3.4 Vibrational Up-pumping: A Tool for Prediction ................................................................... 29

1.4 RESEARCH CONCEPT AND AIMS .......................................................................................................... 34

1.5 REFERENCES ................................................................................................................................... 35

EXPERIMENTAL AND COMPUTATIONAL METHODS ........................................................................ 43

2.1 COMPUTATIONAL METHODS ............................................................................................................. 43

2.1.1 The Schrödinger Equation ................................................................................................... 43

2.1.2 Hartree-Fock Theory ........................................................................................................... 45

2.1.3 Multi-Reference Methods ................................................................................................... 48

2.1.4 Density Functional Theory ................................................................................................... 51 2.1.4.1 Hohenberg-Kohn Theorems ......................................................................................................... 51 2.1.4.2 Kohn-Sham Equations................................................................................................................... 52 2.1.4.3 Exchange-Correlation Functionals ................................................................................................ 55

2.1.5 Basis Sets ............................................................................................................................. 57 2.1.5.1 Localised Basis Set – Isolated Molecules ...................................................................................... 58 2.1.5.2 Condensed Matter, Delocalised Basis Sets and Bloch Theorem ................................................... 59 2.1.5.3 Pseudopotentials .......................................................................................................................... 62

2.1.6 Phonon Calculations ............................................................................................................ 64

2.2 EXPERIMENTAL METHODS ................................................................................................................. 68

2.2.1 X-ray Diffraction .................................................................................................................. 68 2.2.1.1 X-ray Powder Diffraction .............................................................................................................. 70

xiv

2.2.2 Inelastic Neutron Scattering Spectroscopy ......................................................................... 72 2.2.2.1 Generation of Neutrons................................................................................................................ 74 2.2.2.2 The TOSCA Instrument ................................................................................................................. 75 2.2.2.3 Neutron Scattering ...................................................................................................................... 76

2.2.3 BAM fall Hammer ................................................................................................................ 79

2.3 REFERENCES ................................................................................................................................... 82

VIBRATIONAL UP-PUMPING: PREDICTING IMPACT SENSITIVITY OF SOME ENERGETIC AZIDES ....... 87

3.1 INTRODUCTION ............................................................................................................................... 87

3.2 AIMS ............................................................................................................................................. 91

3.3 TEST SET OF ENERGETIC AZIDES .......................................................................................................... 92

3.4 METHODS ...................................................................................................................................... 94

3.5 RESULTS AND DISCUSSION ................................................................................................................. 99

3.5.1 Bond Rupture of Explosophoric N3− ..................................................................................... 99

3.5.1.1 Dissociation of N3− ...................................................................................................................... 103

3.5.2 Metallisation in the Azides: Case Study of 𝛼-NaN3 ........................................................... 109 3.5.2.1 Band gap dependence on external lattice modes in α-NaN3 ...................................................... 114 3.5.2.2 Band gap dependence on internal vibrational modes in α-NaN3 ............................................... 118

3.5.3 Up-Pumping and Impact Sensitivity .................................................................................. 123 3.5.3.1 Partitioning of the Vibrational Structure .................................................................................... 127 3.5.3.2 Coupling Pathways and Impact Sensitivity ................................................................................. 131

3.6 CONCLUSIONS ............................................................................................................................... 140

3.7 SUGGESTIONS FOR FUTURE WORK .................................................................................................... 142

3.8 REFERENCES ................................................................................................................................. 143

VIBRATIONAL UP-PUMPING IN SOME MOLECULAR ENERGETIC MATERIALS ................................ 151

4.1 INTRODUCTION ............................................................................................................................. 151

4.2 AIMS ........................................................................................................................................... 153

4.3 MODEL SYSTEMS ........................................................................................................................... 153

4.4 METHODS .................................................................................................................................... 156

4.5 RESULTS AND DISCUSSION ............................................................................................................... 160

4.5.1 Electronic Structure ........................................................................................................... 160

4.5.2 Vibrational Structure of Some Organic Energetic Materials ............................................. 161

4.5.3 Vibrational Up-Pumping in the Molecular Energetic Materials ........................................ 167 4.5.3.1 Overtone Pathways .................................................................................................................... 173 4.5.3.2 Combination Pathways ............................................................................................................... 178 4.5.3.3 Two-Layer Combination Pathways ............................................................................................. 182 4.5.3.4 Temperature Dependent Up-Pumping ....................................................................................... 185 4.5.3.5 Up-Pumping from Zone-Centre Frequencies .............................................................................. 199

4.6 CONCLUSIONS ............................................................................................................................... 202

4.7 SUGGESTIONS FOR FURTHER WORK .................................................................................................. 204

4.8 REFERENCES ................................................................................................................................. 205

VIBRATIONAL UP-PUMPING IN POLYMORPHIC MATERIALS ......................................................... 210

5.1 INTRODUCTION ............................................................................................................................. 210

5.2 AIMS ........................................................................................................................................... 214

5.3 MATERIALS .................................................................................................................................. 215

5.4 RESULTS AND DISCUSSION ............................................................................................................... 218

5.4.1 Polymorphism of HMX ...................................................................................................... 218

xv

5.4.1.1 Electronic Structure of HMX Polymorphs ................................................................................... 218 5.4.1.2 Vibrational Structure of HMX Polymorphs ................................................................................. 219

5.4.2 Polymorphism of FOX-7..................................................................................................... 224 5.4.2.1 Experimental Impact Sensitivity ................................................................................................. 224 5.4.2.2 Electronic Structure .................................................................................................................... 227 5.4.2.3 Vibrational Up-Pumping in FOX-7 Polymorphs ........................................................................... 227

5.5 CONCLUSIONS ............................................................................................................................... 233

5.6 SUGGESTIONS FOR FURTHER WORK .................................................................................................. 235

5.7 REFERENCES ................................................................................................................................. 235

GENERAL CONCLUSIONS AND FUTURE DIRECTIONS ..................................................................... 239

6.1 GENERAL CONCLUSIONS ................................................................................................................. 239

6.2 FUTURE DIRECTIONS ...................................................................................................................... 245

6.3 REFERENCES ................................................................................................................................. 247

APPENDIX A ................................................................................................................................. 249

APPENDIX B ................................................................................................................................. 251

1

Chapter 1

INTRODUCTION

1.1 Energetic Materials

1.1.1 Energetic Materials: A Brief History

Energetic materials (explosives, propellants and pyrotechnics; EMs) contain

stored chemical energy which is rapidly released on initiation. EMs have

proved to be of great value for industrial, commercial and military applications.1

The origin of EMs is often ascribed to the accidental discovery of black powder

(a mixture of KNO3, S8 and charcoal) ca. 220 BCE in China. Their development

in Europe began much later, when the English monk Roger Bacon began

further studies of black powder in 1249 CE, described in his letter ‘On the

Marvellous Power of Art and Nature and on the Nullity of Magic’.2 This led to

the adoption by western nations of black powder for military applications by

the end of the 13th century. Despite these early discoveries, EMs were not

used industrially until some time later, with the first documented use of black

powder in England in the 1670s.3 Black powder quickly became notorious for

its propensity to accidentally initiate, despite numerous attempts at

desensitising the material using additives, including paraffin and starch.

However, it remained the primary industrial EM until the 1870s.3

It was not until the late 19th century that substantial progress was made on EM

technologies.3,4 The Nobel family made enormous strides with the

development of nitroglycerine (NG) based materials, including a variety of

dynamite compositions.5 A mixture of NG with clay (Guhr dynamite) proved

sufficiently stable for industrial application, and mixtures of NG with

nitrocellulose (NC) led to formation of gelatine dynamite. Both dynamite forms

2

remain in use today. The Nobel family also introduced mercury fulminate as

an alternative detonator to replace black powder. Ammonium nitrate also

became a popular additive to enhance explosive compositions.

Throughout the development of EMs, safety remained a top concern.

Accidental, explosive-related casualties remained very high across industry,

and it was finally recognised that government-regulated standards should be

imposed to ensure explosives were fit for purpose. Both dynamite and black

powder were barred from use, and ammonium nitrate-based compositions

became favoured for industrial application.3

Figure 1.1: Chemical structure diagrams for common molecular energetic compounds.

Alongside the development of industrial EMs was the production of new

military-grade materials. Picric acid, Figure 1.1, was an early favourite towards

the end of the 19th century. However, when loaded into munitions, it had a

tendency to react with the metallic shell walls, leading to highly sensitive metal

salts.6 This was largely overcome at the turn of the 20th century with the

introduction of trinitrotoluene (TNT), Figure 1.1. This became a widely used

explosive during the first World War. A number of other EMs were developed

prior to the second World War, including pentaerythritoltetranitrate (PETN),

1,3,5,7-tetranitro-1,3,5,7-tetrazocane (Octogen or HMX) and 1,3,5-

Trinitroperhydro-1,3,5-triazine (RDX). The latter two were favoured for military

use due to their lower impact sensitivity, Figure 1.1.

The sensitivity of RDX and HMX remained a problem, and safety concerns

persisted. Rather than developing new EMs, it was instead found that the

3

sensitivity of energetic crystals could be reduced by embedding into polymer

matrices to produce polymer-bonded explosives (PBXs). Semtex is a well-

known PBX containing PETN and RDX, although very many PBXs are known

and used today.7

1.1.2 Insensitive Munitions

While the early development of EMs was slow, the latter half of the 20th century

has seen rapid development of many new EMs. Generally, new molecules are

desired that can be more safely handled and which exhibit more powerful

energetics properties.8,9 The need for safe EMs was finally recognised globally

in the 1970s with the establishment of outlines for insensitive munitions (IMs).10

An IM describes any EM that will not initiate under any condition other than its

intended use, and has been made to include explosive formulations (e.g. PBXs)

as well as insensitive explosive molecules. For example, HMX and RDX only

meet IM regulations when formulated as PBXs. A variety of replacement EMs

have been proposed, Figure 1.2. For example, hexanitrostilbene (HNS) and

triaminotrinitrobenzene (TATB) are insensitive materials which exhibit high

thermal stability. Nitro-1,2,4-triazol-3-one (NTO) is a high-energy, low-

sensitivity material that has been suggested as a replacement for TNT and has

found commercial applications (e.g. in automobile airbags). 1,1-diamino-2,2-

dinitroethylene (FOX-7) has also become a popular EM, exhibiting excellent

energetic properties and low sensitivity.

Figure 1.2: Molecular structure of new insensitive EMs.

An alternative to developing new molecules has been to generate multi-

component materials: co-crystals and salts. The potential to tune EM

properties by multi-component crystallisation was noted early by T. Brill, who,

4

while studying solvates of HMX, noted that ‘the physical and chemical

properties of HMX might be tailored systematically by such dopants’.11 Very

many examples of multi-component EM crystals are now known (with both

energetic and non-energetic co-formers), and in many cases exhibit drastically

different sensitivity properties when compared to the pure EM.12,13 Notable

examples include co-crystals of TNT14, including a co-crystal with the highly

sensitive EM hexanitrohexaazaisowurtzitane (CL-20 or HNIW), CL-20●TNT.14

This co-crystal exhibits substantially reduced impact sensitivity as compared

to either pure EM. A number of co-crystals are also known based on HMX15,

including HMX●2-bromoaniline and HMX●2-pyrrolidone and 2CL-20●HMX.16

All of these co-crystals exhibit notably different impact sensitivities than pure

HMX. However, despite the possibility to develop new molecules and multi-

component crystals, there remains very limited understanding of what

constitutes a sensitive EM. Hence any new EM requires synthesis and

thorough testing, at great cost and risk to safety.

Despite the enormous libraries of known explosive materials, safety and

performance remain of utmost importance. The testing required to validate the

safety and performance of new EMs is extensive. As such, very few of these

new candidate molecules make their way into practical application. Instead, it

has been more common to utilize EMs with well-characterised safety

parameters, and vary the composition to which they are added.

1.1.3 Energetic Materials: Definitions and Classifications

Explosions caused by EMs are chemical explosions and are the result of a

rapid chemical reaction that releases large amounts of energy and gas. The

chemical transformation occurs so rapidly that gas products do not instantly

expand out of the reaction zone. This leads to immense pressurisation within

the material and formation of a shock wave.17

An EM can be any material that contains both a fuel and an oxidiser. These

can be single component (e.g. TNT or HMX), or multi-component (e.g. black

powder) systems. When the material ignites and reacts without the formation

5

of additional pressure (i.e. a slow burn), it is said to combust. If, however,

pressure is generated in the material, the material is instead said to deflagrate.

A deflagration is characterised by a sub-sonic burn rate. Under very particular

circumstances – pressure build-up, confinement or very rapid reaction –

deflagration can instead change to a detonation. This transition, known as the

deflagration-to-detonation (DDT) transition, results from adiabatic heating,

i.e. that that temperature increases with pressure. Hence, if sufficient pressure

is accumulated within the material during its burn, or an intense shock is

applied to the material, the propagation of the chemical reaction front

accelerates due to increased temperature. In a detonation, the reaction front

propagates at supersonic speeds, and is associated with the propagation of a

shock front.

Figure 1.3: Schematic representation of the structure of a detonation front in an energetic

material. Unreacted material has pressure of 𝑝0, and the shock front is exposed to a pressure, 𝑝1.

The Chapman-Jouguet plane is indicated as 𝑅𝐶𝐽. Figure adapted from Ref. 4

During the propagation of a shock front into the unreacted material, a thin layer

(ca. 10-100 Å)18 of material is compressed, Figure 1.3. The pressure

associated with the shock front leads to an increase in temperature of the

material, and initiation of the reaction. As the shock front passes, the pressure

6

(and temperature) of the material behind the front decrease along the shock

adiabat (Hugoniot adiabat). At a characteristic point, the Chapman-Jouguet

Point ( 𝑅𝐶𝐽 ), the chemical reaction reaches an equilibrium, the shock

propagation reaches Mach 1 and the detonation process stops. The rate of

shock propagation depends on the rate at which the chemical reaction can

occur, with typical values of 1500-9000 ms-1.19 Hence, any model aimed at

predicting this phenomenon must include processes no slower than this. The

general scheme for explosion can therefore be summarised in Figure 1.4.

Figure 1.4: Stages of an explosion in an energetic material.

EMs can be classified based on either their structural type or their properties.

In the first approach, molecules which are themselves classified as explosives

contain chemical moieties with explosive properties, known as explosophoric

groups. These explosophoric groups are then used to define the class of an

EM. Plets20 suggested structural groupings of explosives based on eight

structure types, Table 1.1.

7

Table 1.1: Structural classifications of explosive compounds, according to Plets. 20

Group Explosive compounds

-O-O or -O-O-O Inorganic and organic peroxides and ozonides

-OClO2 and -OClO3 Inorganic and organic chlorates and

perchlorates

-N-X2 Where X is a halogen

-NO2 and -ONO2 Inorganic and organic compounds

-N=N- or -N=N=N- Inorganic and organic azides

-N=C Fulminates

-C≡C- Acetylene and metal acetylides

M-C Molecules containing metal-carbon interactions

While these criteria can be useful in the design of new EMs, they do not provide

much insight into their characteristic properties. In addition to discussion of

their chemical properties, it is therefore common to classify EMs by their

physical properties, Figure 1.5. At the highest level, a high explosive is taken

as a material capable of detonating, while a low explosive cannot. Materials

such as propellants and pyrotechnics burn, rather than explode. The

classification of low explosive compounds generally depends on their

applications. Broadly, propellants burn with timescales in the order of

milliseconds, releasing a steady stream of gas and can therefore be used to

generate thrust. Pyrotechnics burn with the intense emission of visible light.

8

Figure 1.5: Classification of energetic materials.

Primary explosives undergo a very rapid transition from burning to detonation.

The explosion products are accompanied with an enormous release of energy,

which can in turn initiate a second, less sensitive EM. These materials will

initiate under mild perturbations. Secondary explosives differ from primary

explosives in that they cannot be readily detonated by heat, impact or friction.

Initiation to detonation requires a shock produced by a primary explosive.

1.2 Initiation of Energetic Materials

1.2.1 Hot Spot Models

The general mode of events in an EM follows the sequence shown in Figure

1.6. The initial mechanical stimulus induces some macroscopic effect. This can

include fracture, shear, plastic deformation, gas pressurisation or another

similar phenomenon. This has the effect of producing microstructural defects

within the material, and concentrates energy in these areas. These

microstructural defects ultimately convert the mechanical energy into heat by

some physicochemical mechanism, leading to a chemical reaction and further

Energetic Material

High Explosive

Primary Explosive

e.g. Lead azide, lead styphnate

Secondary Explosive

e.g. HMX, RDX, TNT,

Low Explosive

Propellant Pyrotechnics

9

heat generation. Finally, dissociated atoms recombine to propagate the

reaction front, leading to release of gaseous products.21

The initiation step results from a large accumulation of energy at the molecular

scale, leading to decomposition. This leads to a chain of reactions and self-

sustaining combustion. The rate at which temperature rises begins to increase,

and leads to deflagration and potentially detonation.

Figure 1.6: Progress of an explosion in a high explosive material. Reproduced from Ref 21.

Early work by Bowden and Yoffe22,23 demonstrated that impacts known to

induce initiation of explosives were associated with bulk heating too low to

allow reaction. This led to the concept of ‘hot spot’ initiation, which remains

popular today.24 They demonstrated that over very small areas (0.1-10 μm),

short mechanical pulses (< 1 ms) could lead to local temperatures of > 700 K.

If hot-spots were outside these parameters, initiation could not occur.22 Hot-

spots with dimensions < ca. 0.1 μm or much lower temperatures may introduce

some decomposition, but quench too rapidly for sustained reaction. Similar

temperatures have also been recorded at the tip of propagating cracks during

fracture. A variety of mechanisms have been proposed25,26 for the formation of

hot-spots, but the mechanism by which they occur depends critically on the

10

nature of the material. Some hot-spots (critical hot-spots) will lead to self-

propagating initiation of an EM, while others (non-critical hot-spots) will lead to

local heating, without initiation of the material. Hence, within the hot-spot

theory for EM initiation, understanding the corresponding hot-spot mechanism

is crucial. Field26 studied the initiation events by high-speed photography, and

concluded that there are only a few mechanisms that are responsible for critical

hot-spot formation as a result of mechanical perturbation. In solids, these are

adiabatic gas heating, friction, adiabatic shear and viscoplastic work.26

Within an explosive composition, gases trapped within defects (on the order of

0.1-10 μm) are adiabatically compressed from an initial pressure, 𝑃𝑖, to some

final pressure, 𝑃𝑓. The temperature of this gas rises according to23

𝑇𝑓 = 𝑇𝑖 (𝑃𝑓

𝑃𝑖)γ−1 / γ

Equation 1.1

where 𝑇𝑖 is the initial temperature, 𝑇𝑓 is the final temperature and γ represents

the specific heat capacities of the phase. Numerical calculations have

suggested hot-spot temperatures to rise in excess of 700-1000 K.27

Frictional heating, resulting from the interaction of explosive particles or with

grit, has also been suggested as an important hot-spot forming mechanism.

The maximum temperature is determined by the lowest melting component at

the contact. In accordance with this effect, Bowden and Gurton28 demonstrated

a method to indirectly measure hot-spot temperatures due to friction by using

grits with different melting temperatures. They determined 700 K to be the

lowest hot-spot temperature required for initiation of PETN (the most sensitive

secondary explosive in common use). The physical base for frictional heating

remains largely debated, but has been suggested to result from accumulation

of stress at the contact surfaces.29 Frictional heating is largely an equilibrium

phenomenon, however, and ignition temperatures tend to be much higher than

the melting temperatures of typical explosive materials. Friction can therefore

lead to melting and local decomposition.30 In many cases it is therefore

11

believed that frictional heating acts in concert with other hotspot

mechanisms.25

Solid compositions contain void space. As a mechanical force is imparted into

a system, material is forced into these voids and are necessarily plastically

deformed. This has been suggested as the principal mechanism for hot-spot

formation in many materials.31

The final principal mechanism for hot-spot formation in solids is via localised

adiabatic shear. This phenomenon stems from the anisotropic deformation of

materials that are exposed to impact or shock. Plastic deformation can localize

into bands, on the order of > 1μm. This is generally the case when thermal

softening exceeds work hardening in a material. In such cases, deformation in

a plane leads to further deformation in the same plane and thus a build-up of

heat.31 This phenomenon was first described by Recht for metals,32 but was

subsequently observed in inorganic explosives by Winter and Field33 and later

in organic explosives, PETN and HMX.26

A microscopic hot-spot model has also been suggested. This model is based

on the concept of dislocation pile-ups.29 Upon mechanical stimulation, the

contact layer undergoes immense plastic deformation and generation of

extended defects (dislocations) that extend into the bulk.29 At any temperature

𝑇 > 0 K, these defects rapidly migrate through the sample and collide

(generally at existing defects), leading to local accumulations of energy within

a crystallite. However, these pile-ups occur over length scales of 10s of

nanometres.34 Hence this mechanism does not produce sufficiently large hot-

spots, and the accumulated energy quickly dissipates to the surrounding bulk.

Hence dislocation pile-ups have been suggested as a non-critical hot-spot

phenomenon.

1.2.2 Vibrational Up-Pumping

While the hot-spot mechanisms describe the generation of large amounts of

energy in localised areas, they do not go so far as to describe localisation of

this energy into a molecular response. An additional model, dubbed vibrational

12

up-pumping was therefore proposed by Coffey and Toton35 in an attempt to

describe the processes occurring immediately behind a shock front. This

model was subsequently developed by Dlott and Fayer.36

The process of vibrational cooling was well established both experimentally

and theoretically through the late 20th century.37 This phenomenon describes

the mechanism by which excess molecular vibrational energy relaxes within a

crystal. However, when mechanical energy is inserted instead into the low

frequency vibrational modes, the reverse process is observed.

Vibrational modes are inherently anharmonic, and the potential energy term of

the Hamiltonian takes the form36

𝑉 = 1/2 ∑𝜕2𝑉({𝜑})

𝜕2𝜑𝜑

𝜑2 + 1/3! ∑𝜕3𝑉({𝜑})

𝜕 𝜑𝜕 𝜑′𝜕 𝜑′′× 𝜑𝜑′𝜑′′

𝜑𝜑′𝜑′′

+. ..

Equation 1.2

where 𝑉(𝜑) is the potential energy surface of the solid, and { 𝜑} is a full set of

normal coordinates, 𝜑 . Previous work demonstrated that in solids, where

displacements are small compared to intermolecular distances, truncation of

𝑉 after the cubic term is valid.38 Hence, noting that mechanical perturbation

directly excites phonon states in a crystal (mainly acoustic modes),35 this

model describes a process whereby the excited phonon state could transfer

energy to an internal vibrational mode by coupling to a third normal mode with

intermediate frequency: vibrational up-pumping.

The rate at which this up-pumping occurs between any set of three modes

depends on the strength of their anharmonic coupling, i.e. the second term of

𝑉 . Because the intermolecular potential is more anharmonic than the

intramolecular potential, coupling processes that include higher numbers of

external modes (q) are dominant over processes that contain higher numbers

of internal modes (Q). Calculations on naphthalene39 suggested the coupling

to decrease by an order of magnitude with inclusion of Q terms, hence 𝑞1𝑞2𝑞3

> 𝑄1𝑞2𝑞3 > 𝑄1𝑄2𝑞3. It follows that upon mechanical perturbation, the phonon

13

bath becomes excited, and equilibrates quickly (in the order of ps).36 This leads

to the formation of a vibrationally ‘hot’ phonon bath and a vibrationally ‘cold’

internal molecular manifold. This state of quasi-equilibrium evolves, with

energy flowing upwards at rates in the order of 10s of ps. Hence, this model

suggests an ability for energy transfer and localisation immediately behind a

shock front,40 and is consistent with prevailing theories of deflagration and

detonation. These theories require primary decomposition reactions to occur

on the time scale of ps.41

In the initial model proposed by Coffey and Toton,35 a direct phonon up-

conversion mechanism was proposed for RDX. Using a complete quantum

mechanical model, the rate of energy transfer from a shock-excited phonon

bath into a select vibrational mode in RDX was calculated. It was demonstrated

that the localisation of energy due to up-pumping was sufficient to overcome

the bond dissociation limit from a mild shock. This result was a crucial step in

understanding localisation of shock energy and hot-spot formation.

The subsequent models proposed by Dlott36,37,39,40 and colleagues instead

suggested an indirect phonon up-pumping mechanism. Using heat flow

models, they calculated the rate of energy up-pumping into the internal

vibrational region. The initial model of Dlott and Fayer36 considered only the

excitation of so-called doorway modes (i.e. modes with frequencies less than

twice the highest phonon frequency). However, subsequent models later

included the effects of doorway mode up-pumping.39 In these models, it was

found that up-pumping occurs in three stages: (1) equilibration of phonon

modes within a time period of < 2 ps, (2) excitation of doorway modes, and (3)

up-pumping of doorway modes only a few ps later. Hence, while the rate-

limiting step is indeed excitation of the doorway modes, additional up-pumping

occurs almost immediately afterwards. Additional work by Toton42 and Bardo43

also discussed the addition of shock pressure in models of nitromethane,

where the pressure response of the vibrational density of states led to changes

in reaction rates according to the up-pumping model.

14

Both Coffey35 and Dlott36,40 noted a particularly intriguing feature of this model.

Defect sites within the crystalline lattice introduce points of extreme vibrational

anharmonicity. Hence, the strength of anharmonic coupling in the vicinity of a

defect is larger and the corresponding rate of up-pumping to these sites is

greater, Figure 1.7. This offered a mechanism for the localisation of energy

near defect sites, and thus the role of internal defects in generating hot-

spots.25,26 However, the exact anharmonic enhancement introduced by defect

sites remains unknown.

Hence, with these early fundamental developments, vibrational up-pumping

appeared to offer a complete mechanism for the introduction, propagation and

localisation of energy, capable of describing initiation of EMs. The

phenomenon has been validated both experimentally and based on theoretical

molecular dynamics simulations.44–47

Figure 1.7: Time dependent vibrational quasi-temperatures for the phonon bath (θ𝑝 ), bulk

vibrational manifold (θ𝑣) and defect sites (θ𝑑). Figure adapted from Ref. 36

15

1.3 Prediction and Rationalisation of Energetic Material Sensitivity

The initiation of energetic materials has been known for centuries. However, a

consistent fundamental mechanism that underpins this phenomenon has not

yet been established. Materials chemistry is now a well-developed field, with

structure-property relations in many materials being thoroughly understood.

Shock-wave physics is also well established. Hence, while tangential

phenomena are largely understood, an understanding of the link between

structure and detonation – i.e. initiation – remains elusive. A detailed

understanding of the factors that govern the sensitivity of a material to the

initiation of chemical reactions that occur under mechanical perturbation has

yet to be obtained. More than most other criteria, the inability to target low-

sensitivity compounds has placed one of the greatest limitations on the

potential for selective and rational development of new energetic technologies.

This problem has attracted considerable attention in recent decades,48–50 with

much interest in discovering the single (or set of) physical parameters that can

be used to rationalise and predict a potential material’s sensitivity. This is a

particularly challenging problem on account of the complex interplay of

chemical and physical events spanning many scales of time and length,51 as

well as the poor consistency of experimentally reported impact sensitivities.52

Correspondingly, many models have been developed to predict and rationalise

the sensitivity properties of EMs. Broadly, these can be grouped into three

approaches, based (1) on the properties of the isolated molecule, (2) on the

properties of the solid state, and (3) on macroscopic descriptors. Typically,

correlations are made against impact sensitivity data reported as the height at

which a drop hammer test will induce initiation with 50% probability, dubbed

ℎ50. For the purpose of the following, discussion will be limited to the main

models that are aimed at predicting sensitivity to mechanical stimulation.

1.3.1 Isolated Molecule Methods

Much of the work that has attempted to rationalise the impact sensitivity of EMs

is based on the study of isolated molecules. Initially, this was a consequence

16

of limited access to crystal structures of these compounds as well as limitations

on computational methods. Numerous models were developed, and the

relative simplicity of models based on isolated molecules keeps their

development an ongoing area of research still today. Broadly, they can be

classified into methods based on structural parameters and those based on

quantum-mechanical properties.

1.3.1.1 Empirical Fitting of Molecular Descriptors

Many attempts have been made to develop fully empirical fits between

molecular descriptors and sensitivities. These methods have been particularly

popular for large-scale screening programmes, and indeed amongst the most

powerful, although they offer no physical mechanism for their success. For

example, Keshavarz53 proposed a general equation based on chemical

composition that was able to fit the impact sensitivities of EM compounds with

formula Ca’ Hb’ Nc’ Od’,

log(ℎ50) = 𝑐1𝑎′ + 𝑐2𝑏

′ + 𝑐3𝑐′ + 𝑐4𝑑

′

Equation 1.3

where 𝑐1 − 𝑐4 are adjustable fitting parameters. It was found that by adjusting

the relative coefficients, a broad range of structural types could be analysed.

However, this fitting remains limited to families of compounds. A variety of

similar equations have also been proposed,54–56 and have generally offered an

excellent and rapid means to assess impact sensitivity of large libraries of

materials.

In a similar fashion, empirical models based on molecular descriptors have

become a popular approach, known as the Quantitative Structure-Property

Relation (QSPR) methods. In these methods, a large number of molecular

descriptors are chosen, including ionisation potentials, electrostatic potentials,

oxygen balance, molecular orbital energies, bond lengths, and many others.

Regression models are subsequently established on large databases and

empirical equations established. Early QSPR-based models employed

17

structural descriptors, with Fayet57,58 being amongst the first to employ

quantum-mechanical descriptors for EMs.

Based on a series of 300 quantum-mechanical molecular descriptors, Fayet58

investigated the impact sensitivity of a total of 161 nitro compounds, separated

into three structural classes, using the QSPR approach. For each class of

materials, this proved very promising, with R2 > 0.8 in each case. The inability

to produce stronger correlations was largely ascribed to poor experimental

data.58,59 A similar approach was also based on 61 non-quantum mechanical

descriptors.59 Other authors have employed considerably smaller numbers of

descriptors. Badder and co-workers,60 for example, built a QSPR model for 10

nitro compounds based on eight quantum-mechanical descriptors, with Shu61

employing as few as two (nitro group charge and oxygen balance) descriptors.

In both cases, reasonable models were obtained for the small test set of nitro-

based compounds studied.

While these empirical methods are promising as screening tools, they offer no

physical insight into sensitivity properties.

1.3.1.2 Oxygen Balance

Based on the assumption that structurally related compounds should undergo

similar decomposition pathways, Kamlet62 and subsequently Kamlet and

Adolph63 proposed a comparison of the impact sensitivity of a compound

against its oxygen balance (OB). The OB was suggested to be relevant as it

describes the ability of a molecule to oxidise itself. That is, compounds that

contain sufficient oxygen to convert all nitrogen to NO2, all carbon to CO2 and

all H to H2O. For C-H-N-O molecules, this is defined as63

𝑂𝐵100 =100(2𝑛𝑜 − 𝑛ℎ − 2𝑛𝑐 − 2𝑛𝑐𝑜𝑜)

𝑀𝑤

Equation 1.4

Across a series of over 70 compounds,62 a logarithmic correlation between the

50% impact heights (ℎ50) and OB. However, the compounds were found to

18

follow two different trends, depending on the structural features. For example,

compounds with the same number and relative position of nitro groups

followed trends, or compounds with/without α-C-H linkage followed their own

trends (see Figure 1.8). This method has since been applied numerous times

in the literature,64 often with very good results. However, these methods are

largely restricted to correlations within structural types, and cannot establish

correlations between these series – even for structurally similar compounds

like TNT and picric acid (see structures in Figure 1.1). If trend lines are mixed,

TNT is predicted to be much more sensitive than Picric acid, Figure 1.8.

Figure 1.8: Correlation of impact sensitivity against OB100 for a series of polynitroaromiatic

compounds. Closed circles are molecules with α-CH linkage (e.g. TNT, #1), and open circles do not

(e.g. picric acid, #2). Figure adapted from Ref. 63

19

1.3.1.3 NMR Chemical Shift

The chemical shifts obtained in NMR are strongly dependent on the electronic

structure of the molecule. It was therefore proposed that these chemical shifts

should reflect the relative bond strength of a structural moiety to the molecule

backbone, and thus give an indication of the stability of the structure.

Correlations were initially made by Owens65 between the 1H NMR chemical

shifts and the impact sensitivity of trinitroarene compounds. A similar model

was subsequently extended based on 15N and 13C NMR chemical shifts by

Zeman.66–68 It has also been extended to the investigation of friction

sensitivities.69 While this has not yet become a widespread approach, it has

demonstrated itself as a powerful method for predicting impact sensitivities of

related compounds, Figure 1.9. However, it is evident that no reliable

information can be obtained by this method if structurally unrelated compounds

are compared.

Figure 1.9: Comparison of impact drop energy (Edr) against 15N NMR chemical shifts of the aza

nitrogen atoms to which -NO2 groups are attached. These are chosen as they are believed to be

involved in the initial step of initiation. Figure from Ref. 67

20

1.3.1.4 Bond Energies and Dissociation

A very popular method to investigate impact sensitivity (as well as thermal

stability) of EMs has been based on the ab initio study of bond dissociation

energies. The first ab initio results to this effect were performed at the Hartree-

Fock level with a STO-3G basis set by Owens et al.70 Owens demonstrated

that the electronic density at the mid-point of the C-NO2 bonds in a series of

EMs correlated well with sensitivity, Figure 1.10A. Similar studies were

performed by other groups, and the correlation substantiated further.71 From

this stemmed additional work in which these -NO2 moieties were

computationally cleaved, and the dissociation barriers hence calculated.72 The

first attempt at comparing these dissociation barriers to impact sensitivity was

suggested by Rice and co-workers and gave promising results, Figure 1.10B.73

This method continues to be a popular means to assess the stability and

sensitivity of EMs. It has been applied to a variety of materials.74,75 However,

a thorough analysis by Mathieu has demonstrated that the correlation of bond

dissociation energies against impact sensitivities only holds across families of

structurally-related compounds.76 Despite its widespread use, the investigation

of bond dissociation, or the concept of the ‘trigger linkage’63 assumes a simple,

single-step decomposition model. Such models have been widely debated,

with both experimental77 and theoretical results77–80 for various molecular

energetic materials suggesting more complex pathways are more likely. Often,

decomposition may instead occur following a series of intramolecular

isomerisation processes, such as C-NO2 → C-O-NO.77–80 In such cases, an

understanding of the dissociation barriers of C-NO2 may be limited in its use.

Hence the physical basis for studying BDEs is limited, although its limits are

not yet known.

21

Figure 1.10: Correlations of C-NO2 bond energies and impact sensitivity. (A) Correlation of the

electron density at the C-NO2 bond mid-point against sensitivity. From Ref 65 (B) Correlation of

C-NO2 bond dissociation energy and impact sensitivity. From Ref 73.

Largely based on the early findings by Owens et al,70 it has been suggested

that much of the information regarding bond dissociation energies can be

obtained from a much simpler calculation: the electrostatic potential.81 This

methodology has been largely pioneered by Politzer and co-workers,82 and

has been applied to molecular and ionic energetic species.83 Qualitative

analysis of the electrostatic potential surfaces has been used to rank impact

sensitivity, noting that molecules with more positive potentials tend to be more

sensitive, Figure 1.11.82,84 This can largely be rationalised by a lower electron

density and hence decreased stabilisation of the molecule. However, Politzer

suggested use of a set of descriptors – the first to describe the average

deviation of the electrostatic potential across a bond, and a second to indicate

the maximum value of the potential.84 The former is taken to describe the

charge separation, and hence the covalency of the bond, with the latter a

measure of the maximum interaction strength, noting that the interaction

energy is proportional to charge density.85–87 Numerous studies have

employed investigation of the electrostatic potential to rationalise impact

sensitivities.88–93 However, many of these investigations tend to conduct such

22

studies on small subsets of molecules, and therefore the wider applicability of

this approach is unknown. However, it is reasonable to assume that it will also

be limited to subsets of molecules that exhibit similar electronic structures.

Figure 1.11: Electrostatic potential surfaces for polynitroaromatic molecules. Surfaces were

calculated at B3LYP/6-31G* level and coloured according to the legend at the top of the figure.

The experimental drop heights (h50) are given below. Figure and values from Ref 88.

1.3.2 Solid State Methods

Despite the progress made in predicting properties of materials from isolated

molecules, such models are limited. For example, description of the isolated

molecule cannot rationalise the effects of polymorphism16,94 or multi-

component crystallisation12 on sensitivity properties. This has led many

researchers to move towards investigating mechanisms based on the

crystalline state.

23

1.3.2.1 Crystal Packing and Non-Covalent Interactions

A number of authors have suggested structural arguments to rationalise

impact sensitivity. The instantaneous, adiabatic compression of a solid leads

to an increase in its final equilibrium temperature. The more compressible is

the material, the higher the final temperature. Politzer and co-workers95,96

therefore suggested that a simple trend for impact sensitivity could be sought

in calculation of the free volume per molecule within the unit cell,

Δ𝑉 = 𝑆/𝑍

Equation 1.5

where Z is the number of molecules and 𝑆 is the free space

𝑆 = 𝑉𝑐𝑒𝑙𝑙(1 − packing coefficient)

Equation 1.6

Very simple in its approach, this method appeared to offer reasonable results,

Figure 1.12. However, these results proved to be highly system dependent,

with different types of EMs following considerably different trends.

The crystalline state is characterised by the type of intermolecular interactions

it contains. This has led many authors to seek sensitivity arguments based on

a study of these intermolecular interactions. Cartwright and Wilkinson97 for

example, suggested that compression of solids leads to formation of new

intermolecular contacts, permitting bimolecular reactions to occur. Their

investigation of a series of inorganic azides therefore focused on correlating

impact sensitivity against the distance between nearest non-bonded nitrogen

atoms. A number of authors have also attempted to correlate the type and

strength of intermolecular interactions with sensitivity,16,93,98,99 with findings

that larger numbers of strong intermolecular interactions tend to reduce the

sensitivity of EMs.

24

Figure 1.12: Experimental impact sensitivity (h50) against free space per molecule in the unit cell,

ΔV. Data are shown for (green) nitramines, (blue) nitroaromatics and (red) other EMs that do not

fit these categories. Figure from Ref 95.

Analysis of the crystal packing arrangements in crystalline materials has been

suggested as an alternative method to rationalise impact sensitivity. Early work

by Coffey100 suggested that the rate of plastic deformation in EMs could be

linked to sensitivity. This was recently developed somewhat tangentially by

Zhang,101 as well as Shreeve and co-workers92 who constructed a model

based on the accumulation of energy due to mechanical strain. They

suggested that studying the deformation potential associated with different

lattice structures could therefore help to rationalise impact sensitivity. For a

pair of multi-component crystals, it was found that packing which included non-

layered components had substantially larger deformation potentials than the

herringbone structure, Figure 1.13A. The material with larger deformation

potential (and hence stored strain energy) was indeed found to be more

sensitive. The same analysis was performed for RDX (a sensitive secondary

25

explosive) and compared to a layered compound, Figure 1.13B.102 Again, it

was shown that in the layered material, lower deformation potentials arose.

Hence a structural mechanism for impact sensitivity was proposed. This has

led to interest in studying the relative strengths of non-covalent interactions,

primarily π… π and hydrogen bonding interactions, which contribute to these

deformation potentials.103,104 A recent study of PETN derivatives has also

suggested that the deformability (e.g. shear or compression) does correlate

well with impact sensitivity.105 While these approaches have proved an

intriguing direction for further research, it has not yet been thoroughly

investigated against a broad range of EMs.

Figure 1.13: Correlation of deformation energy to impact sensitivity. (A) Comparison of

deformation energies in two multi-component materials with different packing arrangements

from Ref 92. (B) Comparison of deformation energies in layered vs non-layered materials, from

Ref. 102

26

1.3.2.2 Electronic Band Gap Criterion and Band Gap Dynamics

Amongst the most popular solid state criteria for assessing impact sensitivity

is the ‘band-gap criterion’.106,107 Noting that bond dissociation requires

population of anti-bonding states, this simple analysis is based on

consideration of the energy gap between the valence and conduction bands.

Within this approach, materials with larger band gaps (i.e. those whose

electronic transitions are less probable) are less sensitive. While extensive

investigation of this band gap criterion is limited, it has been applied with

varying success. Perhaps the largest drawback to this approach is the

unreliable calculation of electronic band gaps within most commonly available

computational methods,108 and the lack of experimental band gap data.

Several authors have expanded this concept to dynamic phenomena. In a

similar spirit to the work of Shreeve and co-workers,92 Kuklja109–111 investigated

the electronic structure of both α-FOX-7 and TATB as a function of different

lattice deformations. Rather than focussing on the deformation potential itself,

Kuklja studied the resulting changes in the electronic band gap and bond

dissociation energy at the interface between sheared planes, Figure 1.14A and

1.14B. Under sufficient shear the band gap of FOX-7 dropped to zero, and the

dissociation energy of -NO2 dropped considerably. In contrast, shear

deformation had no notable influence on the dissociation energy of the -NO2

moieties of TATB,111 Figure 1.14B, although Manaa112 did identify a large

reduction (albeit not to metallisation) in its electronic band gap. This was

suggested as a rationale for the different sensitivities of these compounds.

The effect of shear in α-FOX-7 is particularly noteworthy. The decomposition

of FOX-7 is generally believed to pass via -NO2 → -ONO isomerisation.77 This

renders comparison of -NO2 dissociation energies largely irrelevant (Section

1.3.1.4). However, it was found113 that under shear strain, direct -NO2 scission

at the interface of shear planes becomes more favourable than isomerisation.

Hence, if shear deformation is considered, a comparison of -NO2 dissociation

energies again become important. This offers an excellent example of the

complex interplay of physical and chemical phenomena in the initiation of EMs.

27

Figure 1.14: Effect of shear deformation on the electronic structure of α-FOX-7 and TATB. Figures

adapted from Refs. 110 and 111.

The dynamic nature of electronic band gaps was recently re-examined by

Bondarchuk.114 Again, based on the need to induce electronic excitation,

Bondarchuk investigated the propensity of organic materials to ‘metallise’ (i.e.

reach a band gap of 0 eV) upon compression. Using a combination of particle

shape, Ψ, melting temperature, 𝑇𝑚, the number of electrons per atom, 𝑁𝐹, the

explosive energy content, 𝐸𝑐 , and the metallization pressure, 𝑃𝑡𝑟𝑖𝑔𝑔 ,

experimental impact sensitivity was fit to a so-called sensitivity function,

Ω =Ψ𝑇𝑚

2

𝑁𝐹7 𝑒𝑥𝑝(𝑃𝑡𝑟𝑖𝑔𝑔/1000)𝑒𝑥𝑝(𝐸𝑐/1000)

Equation 1.7

This led to a relatively good correlation against experimental results (R2=0.83).

However, despite the seemingly good correlation, this method offers no real

physical rationale for sensitivity.

1.3.3 Kinetic Models

A relatively new approach to the study of impact sensitivity is based on kinetic

considerations. Pioneered by Mathieu,76,115 these models assume that impact

sensitivity is proportional to the rate of propagation of the initial decomposition

step, i.e. X-NO2 bond scission for nitro-containing compounds. If propagation

28

is too slow, localised energy dissipates away from the reactive sites, and self-

sustained decomposition does not occur. This model states that the impact

sensitivity (ℎ50) is given by116,117

ℎ50 = (kc/𝑘𝑝𝑟 )

𝑛

Equation 1.8

where 𝑘𝑝𝑟 is the rate constant for the propagation of the primary

decomposition pathway, 𝑘𝑐 is a fitted parameter, and n is the order of the

reaction and must be > 0. The rate constant is subsequently constructed as a

function of the number of atoms in a molecule 𝑁𝐴, bond dissociation energies,

𝐷𝑖, the energy released due to decomposition of the first molecule, 𝐸𝑐, and a

set of scaling parameters, c and 𝑍𝑖. This yields

𝑘𝑝𝑟 = 𝑁𝐴−1 ∑𝑍𝑖

𝑖

𝑒𝑥𝑝 (−𝑐𝐷𝑖𝑁𝐴

𝐸𝑐)

Equation 1.9

where the sum is over all possible X-NO2 scission pathways, i, with additional

summation terms required for each identify of X (i.e. O-NO2 vs C-NO2).116

Based on a limited set of input parameters, a QSPR-type regression is

subsequently performed to obtain values of c and 𝑍𝑖. Despite the mathematical

similarity to QSPR methods, the physical basis used in developing this model

has allowed a reduction in the number of required parameters (from hundreds

to only three), and better correlations to large datasets.118

Excellent correlations have been obtained using this approach, with 𝑅2 > 0.8

based on diverse datasets of 93 nitroaliphatic compounds.118 This could be

extended to a larger dataset (156 compounds) including nitroaromatic

compounds with the addition of one extra fitted parameter to reflect an

additional bond type, Figure 1.15.116

29

Based on a physical model of impact-induced reactions, these semi-empirical

methods have proved very powerful for the rationalisation and prediction of

impact sensitivities.

Figure 1.15: Correlation of experimental sensitivity against 𝑘𝑝𝑟. Test set includes a diverse range

of molecules with -NO2 based explosophores. Grey circles include carbonyl moieties. Figure from

Ref 116

1.3.4 Vibrational Up-pumping: A Tool for Prediction

While many of the models discussed above describe decomposition processes,

none are based on the relative rates of energy localisation and subsequent

formation of hot-spots in EMs. To this end, models based on vibrational up-

pumping (Section 1.2.2) have been considered.

Since its conceptual development in the 1980/90s,36,39,40 and experimental

validation44 of the up-pumping phenomenon by Dlott and colleagues, there

30

have been a number of attempts at employing up-pumping models to

predicting impact sensitivity. The initial numerical analyses by Dlott and co-

workers36 were based on rigorous heat-flow models and set out a detailed

understanding of the phenomenon of vibrational up-pumping. These concepts

formed the base for simplified models capable of spanning a range of materials.

Perhaps the first attempt at rationalising sensitivities using an up-pumping

model was that by Fried and Ruggiero.119 In their early model, the vibrational

density of states of a set of energetic materials was generated from inelastic

neutron scattering spectra, Figure 1.16A. A kinetic model for the up-conversion

of energy was developed based on the two-phonon density of states and the

temperature-dependent populations. Despite the very limited quality of data

(limited both by data resolution and the maximum measurable energy transfer),

the resulting trend in predicted sensitivities was very promising, Figure 1.16B.

Importantly, vibrational energy transfer was not considered above 600 cm-1,

although this was primarily the result of the experimental limitations at the time.

A similar approach was taken by Koshi more recently, which included lattice

dynamics calculations of phonon density of states.120

Figure 1.16: Vibrational up-pumping model of Fried and Ruggiero. (A) Phonon density of states

derived from inelastic neutron spectra. (B) Predicted sensitivity ordering based on up-conversion

rates into a select vibrational mode (ω = 425 cm-1) at 300 K. Figures adapted from Ref 119.

31

McNesby and Coffey121 subsequently built a model based on experimental

Raman spectroscopy. In their model, the assumption was made that the rate-

determining step in vibrational up-pumping is the transfer of energy from the

phonon manifold to the doorway region, consistent with prevailing theory. The

phonon bath was arbitrarily defined as modes with 𝜔 < 250 cm-1. Following

on from the work by Fried and Ruggiero,119 all up-pumping into the region with

ω < 700 cm-1 was considered, based on a kinetic analysis of the harmonic

overtones. Following from Fermi’s Golden Rule, overtone modes that were off-

resonance with a doorway mode scattered more slowly. Despite the

assumptions made, this proved promising in predicting the relative ordering of

impact sensitivities, Figure 1.17.

Figure 1.17: Relative rate of energy up-conversion to the doorway region for a series of energetic

materials. Note the exponential trend. Figure from Ref 121

Koshi122 was subsequently first to employ an approach based on ab initio

quantum mechanical calculations, and on a rate equation proposed by Dlott,36

κ =𝑗ℏΩ

𝜏1(0)𝜃𝑒

Equation 1.10

32

Here, 𝑗 is the number of doorway modes with frequency Ω and 0 K lifetime τ1.

θ𝑒 describes a temperature equivalence term, which describes the

temperature at which the rate of transfer into a doorway mode is the same as

the low temperature rate at which it transfers out of this same doorway mode

by two-phonon emission

𝑛Ω/2(θ𝑒) − 𝑛Ω(θ𝑒) = 1

Equation 1.11

Noting that Ω/θ𝑒 is constant, and noting that τ1(0) is approximately constant

for organic EMs (ca. 2-6 ps),40 Koshi argued that the rate of energy transfer

therefore depends only on the number of doorway modes. Noting the

discrepancy in the upper limit of the doorway region in earlier studies, the

doorway region was varied with 2Ω𝑚𝑎𝑥 set at 500, 600 and 700 cm-1 and the

number of doorway modes counted in each material. The gas-phase

frequencies were calculated, and the number of doorway modes correlated

well with experimental impact sensitivities Figure 1.18.

Figure 1.18: Comparison of the number of doorway modes in a series of EMs and experimental

impact sensitivities. Doorway mode frequencies were based on ab initio calculations. Figure from

Ref.122

33

Very recently, Bernstein123 has expanded on this method. The frequencies of

crystalline materials were calculated by ab initio methods, and the harmonic

overtones extrapolate. Bernstein subsequently correlated the number of

overtone frequencies that were ‘near resonant’ (i.e. ω ± 10) with fundamental

doorway modes in the region 200 < ω < 700 cm-1. Again, this led to good

correlation with experiment.

The major drawback to these studies has been an inability to directly calculate

the anharmonic coupling constants, which depend on the material and

associated vibrational frequencies. While it is in principle possible to calculate

these from ab initio methods, computational approaches are currently too

intensive. Most studies have assumed these to be constant for all materials,

while some have attempted to approximate them based on simple force-field

potentials.120 Most recently, McGrane124,125 considered vibrational up-pumping

in HMX, TATB and PETN by extracting the anharmonic potentials from high

resolution Raman spectra. All three materials were found to exhibit similar

average anharmonicities, and therefore lends validation to previous models in

which this term is neglected.

These models have proved very promising. The rather limited application of

these models in the last 20 years can only be ascribed to their difficulty.

Calculation of the vibrational structure is a long, arduous task that has only

recently become computationally feasible. Furthermore, the quality of

spectroscopic data, and in particular inelastic neutron scattering data, has only

reached sufficiently high resolution in recent years.126 Moreover, previous

models have varied largely in their assumptions and no thorough analysis has

yet been undertaken. It is therefore very timely to re-examine this model as a

physical basis from which to understand mechanically-induced reactions in

EMs.

34

1.4 Research Concept and Aims

The development of new EMs is an active area of fundamental research.

Amongst the main target characteristics of new EMs, ensuring low sensitivity

to mechanical perturbation is simultaneously a top priority and notoriously

difficult. This difficulty is largely due to the fact that no underlying mechanism

for EM sensitivity is yet known.

It is generally accepted that initiation of an EM requires localisation of energy

within the material. This localisation can be described by hot-spots. The origin

of these hot-spots remains widely debated, although a number of mechanisms

(Section 1.2) have been proposed. Some theories suggest hot-spots to be

rooted in equilibrium temperature increases, while others describe non-

equilibrium, athermal processes.

Many models (Section 1.3) have been proposed in an attempt to identify a set

of physical parameters to understand and describe the impact sensitivity of

EMs. The ‘band gap criterion’ remains the most popular amongst solid state

models. While many models are promising, they often lack a physical basis

and are restricted to subsets of energetic materials. A particularly promising

physical model is rooted in the so-called up-pumping of vibrational energy

(Section 1.2.2). This model describes the transfer of energy from an initial

mechanical impulse, and the mechanism by which it transitions into localised

molecular energy. Importantly, the up-pumping model is not isolated from

previous hot-spot models, as the initial energy can originate via any

phenomenon, including adiabatic compression, fracture, plastic deformation,

contact stresses, amongst others. Hence, it offers a physical basis for

converting hot-spot generation mechanisms into a chemical reaction.

Considerable experimental work has validated the up-pumping phenomenon,

and numerical modelling has been used to rationalise many of its general

features. However, only very limited work has focused on applying these

concepts to test sets of EMs for the prediction of impact sensitivity. Those that

have done so have been based on low resolution inelastic neutron scattering

35

spectra,119,120 or very limited consideration of vibrational structure from

calculation,123,124 or on Raman spectroscopy.121 The models employed in

these studies have been limited to very simple (and often different) energy-

transfer models and typically require the inclusion of additional experimental

data. Before a model based on up-pumping based model can therefore be

employed to predict impact sensitivity, a fully ab initio model must be

developed.

This work therefore conducted with the following aims:

• Investigate the vibrational properties of a range of energetic materials.

• Consider possible target vibrational modes for simple EMs.

• Consider the ‘band-gap criterion’ for a range of EMs.

• Investigate the development and use of an up-pumping based ab initio

model to predict the relative impact sensitivities of a range of energetic

materials.

• Validate potential models against available experimental impact

sensitivities.

• Unify previously proposed predictive up-pumping models into a single

model.

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(113) Kuklja, M. M.; Rashkeev, S. N. Interplay of Decomposition Mechanisms at Shear-Strain Interface. J. Phys. Chem. C 2009, 113 (1), 17–20.

(114) Bondarchuk, S. V. Quantification of Impact Sensitivity Based on Solid-State Derived Criteria. J. Phys. Chem. A 2018, 122, 5455–5463.

(115) Mathieu, D. Theoretical Shock Sensitivity Index for Explosives. J. Phys. Chem. A 2012, 116 (7), 1794–1800.

(116) Mathieu, D.; Alaime, T. Predicting Impact Sensitivities of Nitro Compounds on the Basis of a Semi-Empirical Rate Constant. J. Phys. Chem. A 2014, 118 (41), 9720–9726.

(117) Mathieu, D.; Alaime, T. Impact Sensitivities of Energetic Materials: Exploring the Limitations of a Model Based Only on Structural Formulas. J. Mol. Graph. Model. 2015, 62 (2), 81–86.

(118) Mathieu, D. Physics-Based Modeling of Chemical Hazards in a Regulatory Framework: Comparison with Quantitative Structure-Property Relationship (QSPR) Methods for Impact Sensitivities. Ind. Eng. Chem. Res. 2016, 55 (27), 7569–7577.

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43

Chapter 2

Experimental and Computational Methods

2.1 Computational Methods

2.1.1 The Schrödinger Equation

The ultimate goal of common quantum chemical approaches is to reach a

(approximate) solution of the time independent, non-relativistic Schrödinger

equation,1,2

��Ψ(𝑥1 , 𝑥2 , . . . 𝑥𝑁 , 𝑅1 , 𝑅2

, . . . 𝑅𝑀 ) = 𝐸Ψ(𝑥1 , 𝑥2 , . . . 𝑥𝑁 , 𝑅1

, 𝑅2 , . . . 𝑅𝑀

)

Equation 2.1

for a system of M nuclei and N electrons, where �� is the Hamilton operator (or

Hamiltonian) and Ψ is the wavefunction that describes the system. Note that

Equation 2.1 combines the electron spatial coordinate ( 𝑟 , xyz) and spin

coordinate (𝑠𝑖 , either α or β) into a single term (𝑥 ), and the nuclear spatial

coordinates are denoted �� . The Hamiltonian (here defined in atomic units) is

a differential operator that describes the total energy,

�� = −1

2∑∇𝑖

2

𝑁

𝑖=1

−1

2∑

1

𝑀𝐴

∇𝑖2

𝑀

𝐴=1

− ∑ ∑𝑍𝐴

𝑟𝑖𝐴

𝑀

𝐴=1

𝑁

𝑖=1

+ ∑∑1

𝑟𝑖𝑗

𝑁

𝑗>𝑖

𝑁

𝑖=1

+ ∑ ∑𝑍𝐴𝑍𝐵

𝑅𝐴𝐵

𝑀

𝐵>1

𝑀

𝐴=1

Equation 2.2

In Equation 2.2, the terms i and j are electron indices, and run over all N

electrons, A and B are the nuclear indices, running over all M nuclei with

charge Z and mass MA. The five terms in Equation 2.2 thus describe:

44

1. Electron kinetic energy

2. Nuclear kinetic energy

3. Attractive electron-nucleus electrostatic interaction

4. Repulsive electron-electron electrostatic interaction

5. Repulsive nuclear-nuclear electrostatic interaction

Terms 3-5 depend on the spatial separation between electron and nuclear

spatial coordinates 𝑟𝑖 and 𝑅𝐴 , respectively; hence 𝑟𝑖𝐴 = |𝑟𝑖 − 𝑅𝐴

|.

The Schrödinger equation therefore ‘simply’ states that using the

mathematical formulation of the Hamiltonian, the energy of a system can be

extracted from its wavefunction. Unfortunately, due to the infamous many-

body problem, the Schrödinger equation cannot be solved exactly for multi-

electron systems, and a variety of approximations must be made.

Arguably the most important approximation in quantum chemistry is the Born-

Oppenheimer approximation (BOA).3 Within the BOA, the significant

difference in mass between a nucleus and an electron is considered (ca. 1800

times greater in 1H, and > 20000 times greater in 14C). The electrons can

therefore be taken as moving in a field of fixed nuclear geometry, hence term

2 of Equation 2.2 disappears and term 5 becomes a constant described by

Coulomb’s law. The Hamiltonian (and wavefunction) can therefore be split into

electronic and nuclear components, to be solved independently. The

electronic Hamiltonian, ��𝑒𝑙𝑒𝑐 becomes,

��𝑒𝑙𝑒𝑐 = −1

2∑∇𝑖

2

𝑁

𝑖=1

− ∑ ∑𝑍𝐴

𝑟𝑖𝐴

𝑀

𝐴=1

𝑁

𝑖=1

+ ∑∑1

𝑟𝑖𝑗

𝑁

𝑗>𝑖

𝑁

𝑖=1

= �� + ��𝑁𝑒 + ��𝑒𝑒

Equation 2.3

The total energy therefore comes from the sum of electronic energy, 𝐸𝑒𝑙𝑒𝑐 and

the constant nuclear repulsion term 𝐸𝑡𝑜𝑡 = 𝐸𝑒𝑙𝑒𝑐 + 𝐸𝑛𝑢𝑐𝑙, where

45

𝐸𝑛𝑢𝑐𝑙 = ∑∑𝑍𝐴𝑍𝐵

𝑅𝐴𝐵

𝑀

𝐵>1

𝑀

𝐴=1

Equation 2.4

This includes all of the energy of the system under fixed nuclear geometry.

Despite the great simplifications obtained via the BOA, no solution to the

many-body problem is achieved. The third term of Equation 2.3, ��𝑒𝑒, cannot

be solved explicitly for multi-electron systems, and elements of terms one and

two are beyond reach. Methods designed to deal with this issue using a

variety of approximations have been developed and are widely employed in

the field of Computational Chemistry.

2.1.2 Hartree-Fock Theory

The simplest ab initio method is the Hartree-Fock (HF) scheme. The HF

method forms the base for nearly all wavefunction based theories. Noting that

the solution of the complete N-electron wavefunction cannot be solved exactly,

HF theory approximates the wavefunction by an antisymmetrised product1 of

N one-electron wavefunctions, χi (𝑥 𝑖 ), known as spin orbitals, their exact

description depends on the basis set used (Section 2.1.3). This product is

known as the Slater determinant,4 Φ𝑆𝐷

Ψ ≈ Φ𝑆𝐷 =1

√𝑁!||

χ1(𝑥 1) χ

2(𝑥 1) … χ

N(𝑥 1)

χ1(𝑥 2) χ

2(𝑥 2) … χ

N(𝑥 2)

⋮ ⋮ ⋱ ⋮χ1(𝑥 𝑁) χ

2(𝑥 𝑁) … χ

N(𝑥 𝑁)

||

Equation 2.5

where the pre-factor ensures the normalization condition that ∫ |Ψ|2𝑑𝜏 = 1.

Due to the use of the antisymmetrised wavefunction in this way, HF theory is

1 Antisymmetrisation is the result of Pauli’s exclusion principle, which dictates that interchange of

any two fermions (e.g. electrons) must result in a change in sign. Hence Ψ(𝑥 1, 𝑥 2, … 𝑥 𝑖 , 𝑥 𝑗 , … , 𝑥 𝑁) =

−Ψ(𝑥 1, 𝑥 2, … 𝑥 𝑗 , 𝑥 𝑖 , … , 𝑥 𝑁)

46

said to include exact exchange. For example, the Slater determinant of a two-

electron system can be formulated according to

Ψ(𝑥 1, 𝑥 2) =1

√(2)(χ1(𝑥 1)χ2(𝑥 2) − χ2(𝑥 1)χ1(𝑥 2))

Equation 2.6

Hence, the two electrons are indistinguishable, and cannot exist in the same

spin-orbital.

Derivation of the HF equations are outside the scope of this work, but

introductory texts can be found in Reference 5. However, it is useful to briefly

introduce the results here, which lead to definitions of electron exchange and

electron correlation.

The energy of the one-electron wavefunction can be found using the Fock

operator, 𝑓; one electron wavefunctions are found to reduce to,

𝑓χ𝑖 = ϵ𝑖χ𝑖

Equation 2.7

where ϵ𝑖 describes the orbital energy of spin-orbital i and

𝑓𝑖 = −1

2∇𝑖

2 − ∑𝑍𝐴

𝑟𝑖𝐴

𝑀

𝑖=1

+ 𝑉𝐻𝐹(𝑖)

Equation 2.8

The Fock operator hence represents the kinetic energy and the electrostatic

interaction between nuclei and electrons. The third term in Equation 2.8, 𝑉𝐻𝐹(𝑖)

is known as the Hartree-Fock potential. This term describes the average

repulsive potential felt by electron i due to interaction with the other 𝑁 − 1

electrons. Hence, the two-electron repulsive operator (term 3 of Equation 2.3)

is reduced to a simple electron operator 𝑉𝐻𝐹(𝑖), in which the electron-electron

repulsion is only accounted for in an average way. HF theory thus introduces

47

so-called average electron correlation 2 . To see this better, it is worth

considering the Hartree-Fock potential in further detail,

𝑉𝐻𝐹(𝑥 1) = ∑(𝐽𝑗(𝑥 1)

𝑁

𝑗

− ��𝑗(𝑥 1))

Equation 2.9

The term 𝐽 , the Coulomb operator, is defined as

��𝑗(𝑥 1) = ∫ |χ𝑗(𝑥 2)|

2 1

𝑟12

𝑑𝑥 2

Equation 2.10

This operator describes the repulsive potential experienced by an electron at

position 𝑥 1 due to the average charge distribution of another electron in a

second spin orbital χ𝑗.3

The second term in Equation 2.9, ��𝑗 describes the exchange contribution of

the HF potential. This term has no classical interpretation, and is due entirely

to the interaction of spin orbitals,

��𝑗(𝑥 1)χ𝑖(𝑥 1) = ∫χ𝑗∗(𝑥 2)

1

𝑟12

χ𝑖(𝑥 2)𝑑𝑥 2χ𝑗(𝑥 1)

Equation 2.11

This term only exists for electrons of like spin, and results from the

antisymmetry of the Slater determinant. Hence, this term is computed without

approximation in HF theory.

2 Electron correlation within HF is taken as the difference in energy between the real system and the HF-derived energy, 𝐸𝑐

𝐻𝐹 = 𝐸0 − 𝐸𝐻𝐹. Under normal bonding conditions, this difference is mainly due to the short-range instantaneous repulsion that occur between electrons (dynamic correlation). In HF, this potential is treated only as an average, and hence is underestimated. Typically, correlation energies are quite small (ca. 0.04 Eh in H2). 3 Note that the Borne interpretation of the wavefunction states that |χ𝑗(𝑥 2)|

2 𝑑𝑥 2 describes the

probability of finding the electron within volume 𝑑𝑥 2.

48

HF theory therefore offers an approach for approximating the solution of the

Schrödinger equation for an N-electron system, by assuming N non-interacting

particles that move in an effective potential, 𝑉𝐻𝐹,

��𝐻𝐹Φ𝑆𝐷 = 𝐸𝐻𝐹0 Φ𝑆𝐷 = ∑𝑓𝑖Φ𝑆𝐷

𝑁

𝑖

= ∑휀𝑖Φ𝑆𝐷

𝑁

𝑖

Equation 2.12

Despite its simplifications, HF is able to reproduce overall system energies to

within ca. 10% of the most accurate computational approaches (i.e. couple

cluster methods). However, the approximations made by neglect of correlation

can lead to issues surrounding calculation of system properties. Numerous

post-HF methods (e.g. Møller-Plesset Perturbation theory) have been

developed, alongside other theories (e.g. Density Functional Theory) in an

attempt to better model the complex electron behaviour.

2.1.3 Multi-Reference Methods

A particular drawback to the neglect of correlation in HF comes in the form of

non-dynamic (or static) correlation – the fact that in some cases a single-

reference wavefunction cannot describe a given electronic state.6 For example,

for the H2 molecule, HF provides a reasonable estimate for the geometry

around the equilibrium geometry. However, if the bond is elongated, the Slater

determinant generates a wavefunction which can be represented as

(𝐻α …𝐻β) + (𝐻β …𝐻α)+ (𝐻− αβ. . . 𝐻+)+( 𝐻+. . . 𝐻− αβ)

Equation 2.13

Thus the HF scheme builds a wavefunction including equal weightings of two

ionic states. This leads to considerable issues in reproducing the asymptotic

limit of H2 dissociation.5

These problems, amongst others, have been approached using

configurational interaction (CI) methods.7 The full theory of these methods is

extensive and can be found in many textbooks, including Reference 5 and 8.

49

In CI methods, the ansatz wavefunction is taken as a linear combination of

determinants (derived from HF), and are weighted by an appropriate

coefficient,

Ψ𝐶𝐼 = 𝑐1ϕ1 + 𝑐2ϕ2 …

Equation 2.14

To expand beyond the HF method, CI methods introduce excited determinants,

in which an electron is permitted to occupy a valence HF orbital. This

introduces extra flexibility to the electrons within the calculation, and reflects

the fact that excited state orbital structures become important at large

perturbations away from equilibrium, and when considering electronic

excitations. This addition therefore offers a means to treat non-dynamic

correlation. Hence, Equation 2.14 can be re-written in terms of the number

excited determinants used and their excitation level, here denoted S (single)

and D (double) excitations (higher order excitations can be used),

Ψ𝐶𝐼 = 𝑐1ϕ𝐻𝐹 + ∑𝑐𝑠ϕ𝑠

𝑆

+ ∑𝑐𝐷ϕ𝐷

𝐷

… = ∑𝑐𝑖ϕ𝑖

𝑖=0

Equation 2.15

To understand how non-dynamical correlation improves the description of H2

dissociation, the nature of the frontier orbitals must first be considered, Figure

2.1.

Figure 2.1: Schematic representation of the frontier orbitals of molecular H2. Figure from Ref. 5

50

The HF method constricts both electrons to the bonding (𝜙1) orbital, whereas

inclusion of the 𝜙2 antibonding orbital in the CI-based approach permits these

electrons to occupy different sides of the nodal plane. Hence, on dissociation,

CI methods have the flexibility to employ a large weighting to the 𝜙2 orbital on

dissociation, and to suppress the contribution of the 𝜙1 ion states. This

therefore eliminates the so-called ‘left-right’ correlation that hindered the HF

approach.

Truncations of Equation 2.15 formed the base for early CI approaches, such

as CIS and CID. As these methods lead to large imbalance of electron

correlation to the minimal excitations used, they are rarely used today. It is

instead common to employ a multi-configurational approach, such as the

complete active space, CASSCF. In these methods, all excitations are

permitted within a defined ‘active’ space, which includes occupied and un-

occupied HF orbitals. For each excitation, the orbitals are re-optimised. If an

appropriate active space is selected, this limits bias in the correlation energy.

Hence, these methods allow for flexibility in electron energies as a result of

changing the electronic configuration. It follows that multi-reference methods

are particularly adept at treating excited states.

While many multi-reference methods do an excellent job at treating non-

dynamic correlation (e.g. CASSCF), treatment of dynamic correlation remains

limited. The multi-reference configurational interaction (MRCI) method,

however, employ a series of multi-reference wave functions, on which

additional CI calculations are performed (and orbitals optimized).9 This is

amongst the best approaches for including dynamic correlation effects.

In this work, the electronic structure of isolated molecules was monitored as a

function of bond perturbations. Hence non-dynamic correlation can prove

problematic. Further, this work was interested in monitoring the relative

energies of the excited states along these perturbations. Thus, it was

particularly appropriate to employ multi-reference CI methods. Given the small

size of the systems studied here, it was possible to perform full MRCI

51

calculations. Hence, both dynamic and non-dynamic correlation was

accounted for. This was done using the Molpro 2012 software10 in this work.

2.1.4 Density Functional Theory

An alternative approach to handling the Schrödinger equation is Density

Functional Theory (DFT). In contrast to HF-based methods, DFT is not based

on explicit calculation of Ψ. Rather, the energy is extracted directly from the

electron density, 𝜌. This is a particularly attractive approach, since the electron

density is experimentally observable, while a wavefunction is not. An excellent

introduction to DFT can be found in Reference 11 and 12, and more detailed

derivations can be found in dedicated texts.5

2.1.4.1 Hohenberg-Kohn Theorems

Rooted in the BOA Hamiltonian, Equation 2.3, the development of DFT began

with the seminal theories proposed by Hohenberg and Kohn: the Hohenberg-

Kohn Theorems (HK). According to the HK theorems:13

1. The external potential, 𝑽𝒆𝒙𝒕(�� ) is uniquely defined by 𝝆(�� )

Since the Hamiltonian is fixed by this term, Equation 2.3, this suggests that the

energy of the system is uniquely defined by 𝜌(𝑟 ). In addition, this posits that

the 3N spatial coordinates required to define a system in the HF equations can

be reduced to only 3 spatial coordinates.

Conceptually, this theorem stems from the fact that the electron density

(whose integral is the total number of electrons in the system) depends on the

number, charge and position of nuclei. It can be shown14 that because the total

energy is a function of the electron density, so too must its components, and

thus Equation 2.3 can be recast as,

𝐸0[𝜌0] = 𝑇[𝜌0] + 𝐸𝑒𝑒[𝜌0] + 𝐸𝑁𝑒[𝜌0]

Equation 2.16

52

This equation is conveniently separated into the system-dependent term

(𝐸𝑁𝑒[𝜌0] = ∫ 𝜌0(𝑟 )𝑉𝑁𝑒 𝑑𝑟 ) and the terms which are universal, (i.e. their form is

independent of N, RA and ZA),

𝐹𝐻𝐾[𝜌] = 𝑇[𝜌] + 𝐸𝑒𝑒[𝜌] = 𝑇[ρ] + 𝐽[ρ] + 𝐸𝑛𝑐𝑙[ρ]

Equation 2.17

where the electron-electron energy is decomposed into the Coulombic

component (𝐽) and a non-classical component, 𝐸𝑛𝑐𝑙, which includes correlation

and exchange effects. And thus the total energy is defined by

𝐸0[𝜌0] = 𝐹𝐻𝐾[𝜌] + ∫𝜌0(𝑟 )𝑉𝑁𝑒 𝑑𝑟

Equation 2.18

Hence, it appears that the first HK theorem offers a direct link between density

and energy. Despite the immense simplicity of these equations, the problem

again arises that, due to the effects of electron correlation and exchange, no

explicit form for 𝑇[𝜌] or 𝐸𝑒𝑒[𝜌] are known.

2. Variational Principle

This theorem states that the functional that returns the ground state energy of

a system will deliver the lowest energy only if the input electron density is the

true ground state density. Hence, there should exist a universal function,

��[𝜌(𝒓)], that could be used to obtain the exact ground state density and energy.

The only problem is that its actual form is not known.

2.1.4.2 Kohn-Sham Equations

Whilst the theories proposed by Hohenberg and Kohn suggest the existence

of a universal functional, they offer no means to determine what form this

functional should take. Following from Equation 2.18, Kohn and Sham15 made

further developments to render a more tractable form of DFT which enjoys

widespread use today.

53

Kohn and Sham suggested that as much of Equation 2.18 that could be

calculated explicitly, should be. Hence, they proposed that the kinetic and

potential energy terms should instead be treated in a similar way to that of HF

theory. It was posited that a good first approximation would be to set-up a non-

interacting reference system based on a Slater determinant wavefunction and

an effective local potential, 𝑉𝑠(𝑟 ),

��𝑆 = −1

2∑∇𝑖

2

𝑁

𝑖

+ ∑V𝑆(𝑟 𝑖)

𝑁

𝑖

Equation 2.19

The Slater determinant, spin orbitals and one-electron operators are

analogous to the HF cases in Equations 2.5, 2.7 and 2.8, respectively, the only

difference being that the orbitals within the Kohn-Sham approach (the KS

orbitals, 𝜑𝑖) are largely artificial. The value of 𝑉𝑠 must be chosen such that KS

orbitals reproduce the electron density of the real interacting system.4 As was

the case with HF theory, the kinetic energy of the non-interacting system can

therefore be written as

𝑇𝑠 = −1

2∑⟨𝜑

𝑖|∇

2|𝜑𝑖⟩

𝑁

𝑖

Equation 2.20

This term allows calculation of a large subset of the kinetic energy, but will not

be the same as that of the true, interacting system, even for the same electron

density. Hence, the functional 𝐹[ρ] is separated,

𝐹[𝜌(𝑟 )] = 𝑇𝑆[𝜌(𝑟 )] + 𝐽[𝜌(𝑟 )] + 𝐸𝑋𝐶[𝜌(𝑟 )]

Equation 2.21

4 ∑ ∑ |𝜑𝑖(�� , 𝑠)|2= 𝜌𝑜(�� )𝑠

𝑁𝑖

54

where the term 𝐸𝑋𝐶 contains all of the non-classical (interacting) terms that are

neglected in solving for a non-interacting system. This is known as the

exchange-correlation energy,

𝐸𝑋𝐶[𝜌] = (𝑇[𝜌] − 𝑇𝑆[𝜌]) + (𝐸𝑒𝑒[𝜌] − 𝐽[𝜌])

Equation 2.22

Hence, the exchange-correlation energy contains all parts of the total energy

equation that are unknown, including effects of self-interaction, exchange,

correlation and components of the kinetic energy. The total energy therefore

becomes

𝐸[𝜌(𝑟 )] = 𝑇𝑆[𝜌] + 𝐽[𝜌] + 𝐸𝑋𝐶[𝜌] + 𝐸𝑁𝑒[𝜌]

Equation 2.23

The final difficulty was therefore to establish a means to define a unique set of

KS orbitals that represent the non-interacting system. It turns out that this can

be done by application of the variational principle,5 and leads to the so-called

(one-electron) Kohn-Sham equation,

Equation 2.24

Similar to the HF one-electron equations, this must be solved iteratively. It is

also worth noting the similarity of the Kohn-Sham equation with the HF

equation (Equation 2.8), with the main difference being the nature of the spin-

orbitals and the exchange-correlation terms. In Equation 2.24, the only term

that remains unknown is 𝑉𝑋𝐶 (the exchange-correlation potential), which is

defined as,

55

𝑉𝑋𝐶 =δ𝐸𝑋𝐶

δρ

Equation 2.25

Thus, if the form of 𝑉𝑋𝐶 were known, the KS approach would lead to an exact

solution of the Schrödinger equation.

2.1.4.3 Exchange-Correlation Functionals

The most active area of research in DFT development surrounds developing

approximate forms for 𝑉𝑋𝐶. A number of approaches have been made, with

varying complexities. A full discussion of these approaches is outside the

scope of this thesis, but an excellent introduction is provided in Reference 16

and 17. The most basic form of 𝑉𝑋𝐶 is based on the work of Thomas and

Fermi18,19 and known as the Local Density Approximation (LDA).15 It makes

the assumption that the electron density can be treated as a uniform electron

gas and thus the exchange-correlation at a point 𝒓 with density 𝜌(𝒓) should be

the same as that of a uniform gas of the same density.20 Physically, this is

similar to the electronic structure of solid metals, for which LDA works quite

well. However, when molecular solids are considered, this approximation

becomes rather poor. Electrons in such systems are not delocalised, but are

confined to the spaces occupied by the molecules. Therefore, for molecular

systems like those studied in this thesis, the Generalised Gradient

Approximation (GGA) becomes more appropriate. The GGA

approximation21,22 accounts for rapidly changing properties of the electron

density by considering both the charge density at a point, and the gradient of

the charge density to account for local deviations. The most common GGA

functionals are the PBE23 (Perdew, Burke and Ernzerhof), PW9124,25 (Perdew

and Wang) and BLYP (Beck 88 exchange functional26 with the correlation

functional of Lee, Yang and Parr27). PBE is particularly popular for modelling

of molecular crystals, and has been previously demonstrated to perform well

for structural and vibrational properties of molecular energetic materials.28–31 It

has therefore been used in this thesis.

56

In contrast to HF theory, DFT does not treat exchange explicitly, but rather

approximates both correlation and exchange. To rectify this, hybrid DFT

functionals are used. These functionals work by introducing a component of

exact HF exchange into the functional. The amount of HF that is included is

based on substantial paramaterisation against experimental data, and many

hybrid DFT functionals are developed ‘for purpose’, and on a specific class of

compounds. The most common hybrid functional, B3LYP,11 is obtained by

adding gradient corrections to the LDA method, the exchange of Becke and

the correlation function of Lee, Yang and Parr. Generally, hybrid functionals

perform very well and are less computationally demanding than wavefunction

methods, particularly for larger systems.

DFT exchange-correlation functionals are inherently local and they therefore

are not capable of accounting for the long-range dynamic correlation that

results in van der Waals interactions. These interactions are vital to the correct

description of molecular materials, such as those studied in this thesis. As such,

a number of empirical and semi-empirical corrections have been developed.

Most notable are those by Tkatchenko and Scheffler (TS)32 and Grimme.33,34

In the popular D2 scheme (Grimme G0633), the dispersion correction takes the

form,

𝐸𝑑𝑖𝑠𝑝 = −1

2∑∑∑

𝐶6𝑖𝑗

𝑟𝑖𝑗,𝐿6 𝑓𝑑,6(𝑟𝑖𝑗,𝐿)

𝐿

𝑁

𝑗=1

𝑁

𝑖=1

Equation 2.26

where 𝑁 is the number of atoms and 𝐿 is a unit cell translation. For 𝐿 = 0, 𝑖 ≠

𝑗. 𝐶6𝑖𝑗 is the dispersion coefficient for atom pair 𝑖𝑗, and 𝑟𝑖𝑗,𝐿 is the distance

between this pair at translation 𝐿.The final term, 𝑓𝑑,6(𝑟𝑖𝑗,𝐿) works to scale the

dispersion correction force-field such to minimize the term when atoms are

within typical bonding distances. In the common D2 scheme, the 𝐶6 term is

empirical, and hence the dispersion is not sensitive to an atom being in a

particular chemical environment. The TS scheme, however, accounts in part

for chemical environment by accounting for changes in an atoms’ charge

57

density. Other DFT functionals have also been developed which attempt to

include non-local correlation explicitly within the ab initio calculation. A

particularly promising non-local correlation functional is rVV10,35 which has

proven to perform very well for structural and vibrational calculations of

molecular materials. Unfortunately, due to the complexity of the non-local

correlation functionals, they become increasingly expensive with the size of

the unit cell.36

A second issue surrounding the intrinsic localised nature of DFT correlation is

that DFT functionals tend to over delocalise electrons due to their intrinsic self-

interaction. 5 This introduces problems with calculation of electronic band gaps,

where typical GGA functionals tend to grossly underestimate these gaps.37

While there is some improvement with the introduction of HF exchange in the

hybrid functionals, there remain problematic effects with long-range HF

exchange, although cancellation of errors in hybrid methods can solve many

of these problems.38 Development of so-called screened hybrid DFT

functionals has therefore appeared. These methods separate the Coulomb

operator within the HF exchange into short- and long-range effects. The

definition of the range (and thus the length-scale of HF exchange) is varied

with an empirically fit parameter. The HSE0639 (Heyd, Scuseria and Ernzerhof)

functional is one such screened hybrid functional, and has proved particularly

promising for the calculation of electronic band gaps. It was therefore selected

for this purpose in this thesis.

2.1.5 Basis Sets

Solution of the Schrödinger equation, Equation 2.2, requires two components,

the Hamiltonian (i.e. how to evaluate the energy) and the wavefunction (i.e.

what to evaluate the energy of). The former is considered within the theories

of HF and DFT, while the latter depends on the so-called basis set.5 The basis

set is a set of functions whose linear combination represents a wavefunction.

Generally, the larger the basis set, the more accurately the wavefunction will

be described. While it is therefore tempting to use large basis sets, a larger

58

number of functions greatly increases the computational cost. Compromises

must therefore be made.

There are two main types of basis sets: localised and delocalised.8 The former

is typically used in the study of isolated molecules, while the latter is more

commonly used for periodic (e.g. crystalline) materials. Both have been utilised

in this thesis.

2.1.5.1 Localised Basis Set – Isolated Molecules

Modelling the electronic structure of an isolated molecule is typically done

using localised functions that are centred on the individual atoms. These

functions form the atomic orbitals, and their linear combination forms the linear

combination of atomic orbitals, molecular orbital method. Early methods

employed Slater-type functions40 (STOs) as they have a similar form to the

hydrogen atom eigenfunctions. However, these functions could not be handled

efficiently by algorithms, and instead Gaussian-type orbitals (GTO) became

popular. While Gaussian functions are more easily handled, they do not have

the necessary cusp at the nucleus, and they decay too rapidly.5,41 As such, in

practice a contraction of Gaussian primitive functions 𝑔𝑗(𝒓), each with an

appropriate weighting coefficient (𝑑 ) is used. As such, a contracted GTO

(CGTO) containing L primitive functions takes the form

χ𝑖(𝑟) = ∑𝑑𝑗𝑔𝑗(𝑟)

𝐿

𝑗=1

Equation 2.27

The larger the number of CGTOs and primitives used to construct a CGTO,

the more accurate will be the model, and the more computationally demanding

will be the calculation. Noting that chemistry typically involves only the valence

electrons, split-valence basis sets are common, where the number of CGTOs

used to model the core orbitals differs from that used for the valence orbitals.

The Pople42 basis set 6-31G, for example, states that six gaussian primitives

are summed to a single CGTO to model the core shells, while two CGTOs are

employed to model the valence region, one composed of three primitives and

59

one of a single primitive function. This offers additional flexibility to the valence

electrons and permits more accurate representation of perturbations to these

electron orbitals. Additional functions can be added to further enhance the

flexibility of these valence states: polarisation and diffuse functions. The former

describes addition of higher angular momentum functions onto an atom (e.g.

a p-orbital onto an s-orbital), and assists in capturing changes in the shape of

electron density on bonding. Diffuse functions add higher principle quantum

number orbitals (e.g. a larger orbital of the same angular momentum).

Polarisation functions are crucial for accurate capture of anions, polarizable

atoms, excited states and long range interactions. A number of different

families of GTOs exist, and differ mainly in their optimization of the primitive

Gaussian exponents. The most common families5 are the Dunning43 and

Pople42 basis sets.

In this thesis, GTOs are used to study bond elongation and excited state

potential energy surfaces of anions. This requires accurate modelling of the

atom-atom bonding interactions, as well as permitting sufficient flexibility to

capture the excited state. Hence in this work both polarization and diffuse

functions are incorporated. However, due to the computational expense of the

calculation, only a limited number of these functions could be used.

2.1.5.2 Condensed Matter, Delocalised Basis Sets and Bloch Theorem

Treating electrons in a solid, which is essentially an infinite array of periodic

unit cells, brings the additional challenge of solving the Schrödinger equation

for an infinite number of electrons. This problem can be solved by considering

Bloch’s theorem, which states that due to the periodicity of a crystalline

material, it is necessary to consider only the electrons that reside within the

primitive unit cell. Bloch’s theorem states that the wavefunction of an electron

within a perfectly periodic potential can be written as44

ψ𝑗,𝒌(𝒓) = 𝑢𝑗,𝒌(𝒓)𝑒𝑖𝒌.𝐫

Equation 2.28

60

In this equation, 𝑢𝑗(𝒓) describes a function with the periodicity of the potential,

such that 𝑢𝑗(𝒓 + 𝒈) = 𝑢𝑗(𝒓), where 𝒈 is the translational length of the crystal.

The term j describes the band index, and k a wave vector within the first

Brillouin zone. In a typical calculation, k is sampled at a discrete number of

points. This is often done using a Monkhorst-Pack grid,45 which is an unbiased

means of selecting a subset of k points to sample based on a rectangular grid

of points, spaced evenly throughout the Brillouin zone. Appropriate dimensions

for this grid require convergence testing on a system-by-system basis,46 and

depends on complexities in the underlying electronic structure.

As 𝑢𝑗 of Equation 2.28 is periodic, it can be expressed as a Fourier series,

𝑢𝑗,𝑘(𝒓) = ∑𝑐𝑗,𝑮𝑒𝑥𝑝(𝑖𝑮 ⋅ 𝒓)

𝐺

Equation 2.29

where G is a reciprocal lattice vector and 𝑐𝑗,𝑮 are expansion coefficients for

each plane wave. It follows that the electron wavefunctions can be constructed

as a linear combination of these plane waves,

𝑢𝑗,𝑘(𝒓) = ∑𝑐𝑗,𝒌+𝑮𝑒𝑥𝑝(𝑖(𝒌 + 𝑮) ⋅ 𝒓

𝑮

Equation 2.30

Similar to the GTO, a DFT calculation therefore aims to minimize the energy

of this function by optimizing the expansion coefficients through self-consistent

field cycles.

Electronic wavefunctions represented by plane waves are fast to compute, and

further, the set of plane waves is universal and does not depend on the position

or type of atoms in the unit cell.5 Hence, in contrast to localised basis sets, all

systems can be treated with the same set of functions. In principle, the plane

wave series is infinite. However, this is clearly not computationally tractable,

and a truncation must be made. This is done by setting an upper energy limit

61

for the plane wave kinetic energies (and thus electron kinetic energies) to be

incorporated in the wavefunction, 𝐸𝑐𝑢𝑡,

𝐸𝑐𝑢𝑡 =ℏ2

2𝑚|𝒌 + 𝑮|2

Equation 2.31

In contrast to GTOs, plane wave basis sets are therefore readily improved,

simply by the addition of more plane waves (i.e. increasing 𝐸𝑐𝑢𝑡 ). An

appropriate 𝐸𝑐𝑢𝑡 must be optimised for each system studied, and depends

intimately on the pseudopotential (described below) and type of atom being

investigated. The ability of plane waves to correctly model atomic structure is

shown in Figure 2.2.

Figure 2.2: The effect of increase the number of plane waves (PW) on the modelled electron

density of a Na atom. From Ref 47.

62

It is also possible (albeit less common) to build Bloch functions of Equation

2.28 based on GTOs. For an N-electron system, this is done by expanding the

unknown crystalline orbitals, i, as a set of m Bloch functions that are

constructed from local atom-centred Gaussian functions, χ,48

ψ𝒌𝑖(𝒓) = 𝑁 ∑𝑎𝑗𝑖(𝒌) (∑χ𝑮,𝑖(𝒓)𝑒𝑥𝑝(𝑖𝒌 ⋅ 𝑮)

𝑮

)

𝑚

𝑗=1

Equation 2.32

where G is again a reciprocal lattice vector, and terms aji are the scaling

coefficients. The main advantage of using GTOs for periodic systems comes

in the study of the electron density, which is more carefully reproduced by

GTOs. Further, the use of GTOs greatly reduces the number of basis functions

used to describe the system and hence are favoured for use with hybrid

functionals, which have explicit consideration of the Fock matrix (and which

scales directly with the number of basis functions employed). Hence the

electronic band structures calculated in this work were performed almost

exclusively using periodic Gaussian-type Bloch functions, along with the

screened hybrid DFT functional, HSE06. The major disadvantage is that there

is no systematic and unique approach to enhancing the basis set quality as

there is with plane waves, and hence the basis sets used in this work were

obtained from previous works in which they were successfully employed in

similar systems.

2.1.5.3 Pseudopotentials

The Pauli exclusion principle2 dictates that higher electronic states are

orthogonal with all states of lower energy. It follows that the electronic

wavefunction becomes highly oscillating in the core region. The expansion of

the electronic wavefunction in a plane wave basis set requires very large

numbers of plane waves to capture this feature. Fortunately, the core electrons

of an atom are only negligibly affected by the chemical environment, and can

be treated as being frozen. It is therefore possible to replace the ionic potential

with a weaker pseudopotential that mimics the screening effect of the core

63

electrons, and which yields the same valence electron wavefunction outside of

the core region, 𝑟 > 𝑟𝑐.49,50 This has the effect of removing the Kohn-Sham

orbitals of the core states, as well as removing all nodes from the valence

pseudo wavefunction for 𝑟 < 𝑟𝑐. Hence, the use of pseudopotentials greatly

reduces the number of plane waves required to reproduce it, Figure 2.3.

Potentials that place 𝑟𝑐 higher are considered ‘softer’ potentials and require

fewer plane waves to model. However, softer potentials also tend to be less

transferrable.

The choice of pseudopotential is not unique, and many approaches exist.

However, all pseudopotentials must obey the simple criteria including:

1. The core charge of the pseudo-wavefunction must be identical to that

of the atomic wavefunction.

2. The eigenvalues of the pseudo-electrons must be the same as in the

atomic wavefunction.

3. The pseudo-wavefunction and its first and second derivatives must be

continuous at 𝒓𝒄

Figure 2.3: The structure of the valence wavefunction as a function of its distance, r, from the

nucleus. Modification of the ionic potential 𝑍/𝑟 by use of a pseudopotential 𝑉𝑝𝑠𝑒𝑢𝑑𝑜 in the region

𝑟 < 𝑟𝑐 leads to a smooth pseudo-wavefunction, Ψ𝑝𝑠𝑒𝑢𝑑𝑜 as compared to the original

wavefunction Ψ𝑍/𝑟. Figure adapted from Ref. 51

64

A variety of pseudopotential types have been developed that satisfy these

conditions. The most commonly employed pseudopotentials are the ultra-soft

pseudopotentials (USPP). These were introduced by Vanderbilt52 to allow the

lowest possible cut-off energies for plane-wave basis sets.

In addition to the above criteria, norm-conserving pseudopotentials53 (NCPP)

can be generated such that the pseudo- and all-electron wavefunctions yield

the same charge density.5 This is done by generating a pseudopotential that

maintains

∫ 𝝋𝑨𝑬∗ (𝒓)𝝋𝑨𝑬

(𝒓)𝒅𝒓

𝒓𝒄

𝟎

= ∫ 𝝋𝒑𝒔∗ (𝒓)𝝋𝒑𝒔

(𝒓)𝒅𝒓

𝒓𝒄

𝟎

Equation 2.33

where 𝜑𝐴𝐸 (𝑟) and 𝜑𝑝𝑠

(𝑟) are the all-electron and pseudopotential

wavefunctions, respectively. This ensures equality of electronic charge both

inside and outside the core region. Because of this, NCPPs ensure accurate

reproduction of the scattering properties of ions and are most easily developed

into DFT (this is particularly true for density functional perturbation theory,

DFPT) codes. The work presented in this thesis therefore employs norm-

conserving pseudopotentials throughout all lattice dynamics calculations. All

pseudopotentials were taken from databases available within the quantum

chemical software: CASTEP (00PBE_OP for C,H,N and O atoms).

2.1.6 Phonon Calculations

There are two approaches to calculating vibrational properties within DFT: the

linear response (also known as density functional perturbation theory, DFPT)

and the finite differences approach. Only DFPT is used in this thesis and will

therefore be discussed briefly here. An extensive review on the subject can be

found in Reference 54 and an excellent introduction to the field in Reference

55. The use of DFPT over finite differences methods in this thesis is due to the

fact that the latter requires use of supercells to calculate the frequencies at

wave vectors away from zone centre. This very quickly becomes

65

computationally intractable for larger, low symmetry systems such as those

used in this work.

It is convenient to begin by considering the total energy of a unit cell composed

of N atoms, with indices 𝑙. Each atom has three degrees of freedom, denoted

as α, and can describe displacement along the x,y and z axes. DFPT is based

on a Taylor (i.e. derivative) series of small perturbations, 𝑢, of the atoms (𝑙) in

the direction α (i.e. x, y and z), such that,

𝐸 = 𝐸0 + ∑𝜕𝐸

𝜕𝒖𝑙,α

. 𝒖𝑙,α

𝑙,𝛼

+1

2∑

𝜕2𝐸

𝜕𝒖𝑙,α𝜕𝒖𝑙′,α′

. 𝒖𝑙,α. 𝒖𝑙′,α′

𝑙,𝛼,𝑙′,𝛼′

+. ..

Equation 2.33

Where 𝐸0 is a constant (referring to the energy of the equilibrium structure) and

is not considered further. At equilibrium geometry, the forces on all atoms are

zero, and hence the first derivative summation disappears. Within the

harmonic approximation, all terms above the second term are ignored. Hence

only the second summation remains. Note that this is akin to Hooke’s classical

law, 𝑈 = 12⁄ 𝑘𝑥2. It is convenient to define the force constant matrix,

Φα,α′𝑙,𝑙′ =

𝜕2𝐸

𝜕𝒖𝑙,α𝜕𝒖𝑙′,α′

Equation 2.34

It is worth noting here that because phonon calculations are based on Equation

2.33, atomic forces must be converged as close to zero as possible such that

the above approximation holds. This requires geometry optimisation of the

structure prior to phonon calculations. Furthermore, as will be discussed below

because the energy is defined by the wavefunction, a well-optimised

wavefunction is also required.

Assuming periodic boundary conditions (i.e. that 𝑢(𝑹 + 𝑁𝒂) = 𝑢(𝑹) where 𝑁𝒂

is an integer translational vector in the crystal), the displacement 𝒖 of

Equations 2.33 and 2.34 can be written as a plane waves,

66

𝑢𝑙,α = 𝛜𝐪𝑒𝑥𝑝(𝑖(𝐪 ⋅ 𝑹 − ω𝑡)

Equation 2.35

Here, 𝛜 is the polarisation vector, which determines the direction in which ions

move, and 𝐪 is the associated phonon wave vector. Inserting Equation 2.35

into the classical equation of motion gives

Dα,α′𝑙,𝑙′ (𝒒)𝛜𝒒,𝛼,𝑙 = ω2𝛜𝒒,𝛼,𝑙

Equation 2.36

This is an eigenvalue equation that links the vibrational frequencies, ω to the

dynamical matrix, defined as

Dα,α′𝑙,𝑙′ (𝒒) =

1

√𝑀𝑙𝑀𝑙′

∑Cα,α′𝑙,𝑙′ (𝒒)

α

=1

√𝑀𝑙𝑀𝑙′

∑Φα,α′𝑙,𝑙′ 𝑒−𝑖𝒒.𝒓𝛂

α

Equation 2.37

where 𝑀𝑙 is the mass of atom 𝑙. That is to say that the dynamical matrix is the

mass-reduced Fourier transform of the real-space force constant matrix.

The base of ab initio lattice dynamics therefore is to obtain the force constant,

Φα,α′𝑙,𝑙′

which, according to Equation 2.34, are the second derivatives of the total

energy. Thus, Φα,α′𝑙,𝑙′

describes a Hessian matrix.

The forces acting on atoms can be determined according to the Hellmann-

Feynman theorem.56,57 This is a central theorem in both geometry optimisation

and phonon calculations. In bra-ket notation the Hellmann-Feynman theorem

relates the derivative of the total energy with respect to a perturbation, λ, to the

expectation value of the Hamiltonian to that same perturbation,54

𝜕𝐸λ

𝜕λ= ⟨𝜑λ|

𝜕��𝜕λ

|𝜑λ⟩

Equation 2.38

67

It follows from Equation 2.38 that for displacement of the ith nucleus, 𝑹𝒊,

𝐹𝐴 = −𝜕𝐸

𝜕𝑹𝐴

= ⟨𝜑 |𝜕��𝜕𝑹𝐴

|𝜑 ⟩

Equation 2.39

Equation 2.39 is known as the electrostatic force theorem, and is used to

minimise ionic positions during geometry optimisation (hence the need for a

good approximation of the wavefunction). For phonon calculations, however,

it is the Hessian of the Born-Oppenheimer energy surface that is required. This

comes from differentiating the Hellmann-Feynman forces (Equation 2.39) with

respect to nuclear coordinates and noting that from the BOA Hamiltonian in

Equation 2.4 the Hamiltonian depends on nuclear coordinates via the electron-

ion interactions (𝑉𝑹(𝒓)) and the charge density, 𝑛(𝒓). Hence,54

𝜕2𝐸(𝑹)

𝜕𝑅𝑖𝜕𝑅𝑗

= −𝜕𝐅𝒊

𝜕𝐑𝒋

= ∫𝜕𝑛𝑅(𝒓)

𝜕𝐑𝒋

𝜕𝑉𝑅(𝒓)

𝜕𝐑𝑰

𝑑𝒓 + ∫𝑛𝑹(𝒓)𝜕2𝑉𝑅(𝒓)

𝜕𝐑𝑰𝜕𝐑𝑱

𝑑𝒓 +𝜕2𝐸𝑁(𝑹)

𝜕𝐑𝑰𝜕𝐑𝑱

Equation 2.40

and the Hessian matrix of the Born-Oppenheimer energy depends on the

ground state electron density, 𝑛𝑹(𝒓) , and its linear response to nuclear

perturbation, ∂𝑛𝑅(𝑟)/ ∂𝑹𝐼 . The final term in Equation 2.40 describes the

second force imposed by one nucleus on another as a function of the

perturbation (i.e. the second derivative of the nuclear-nuclear interaction

energy).

In a DFPT (or linear response) calculation, the Hessian matrix is generated

according to Equation 2.40 for a select subset of wave vectors and the

dynamical matrix attained via Equation 2.37. These can be subsequently

Fourier interpolated onto other wave vectors, and hence a small number of

explicitly calculated dynamical matrices can yield a complete dispersion curve.

This allows access not only to dispersion relations along high symmetry lines,

but also to generation of phonon density of states, which require consideration

of a large number of wave vectors for accurate production. This assumes that

the calculated wave vectors capture the nature of the force constant along the

68

wave vector that is being interpolated. In an attempt to ensure unbiased

sampling of wave vectors in the initial case, wave vector sampling is often done

using a Monkhorst Pack grid.

2.2 Experimental Methods

2.2.1 X-ray Diffraction

The most definitive way to analyse the structure of a solid is via diffraction-

based techniques. Most commonly, diffraction techniques use X-ray radiation,

although neutron- and electron-based techniques are also well-known. An

excellent introduction to the field of X-ray diffraction and crystallography can

be found in Reference 58.

X-rays are a form of electromagnetic radiation, with wavelength ~ 0.1 to 100

Å. Their interaction with matter results from interactions with the electrons of

the material, and the intensity of their scattering is therefore proportional to the

electron density. While X-rays are less sensitive to atom type (and insensitive

to isotopes), they scatter more strongly than neutrons. Thus, for most purposes,

X-ray diffraction tends to be favoured due to faster collection times and higher

quality data as compared to neutron diffraction.

An additional benefit to X-ray diffraction is its availability, with laboratory X-ray

sources now commonplace. In a laboratory source, X-rays are generated by

accelerating an electron into an anode of a characteristic material. Upon

collision, the kinetic energy of the electron is sufficient to eject a core-level

electron from the anode, leaving an unstable vacant core state. An electron in

a higher energy orbital therefore drops into the vacant state, and the excess

energy emitted as an X-ray. Due to the quantized structure of atomic orbitals,

the X-ray energy is characteristic of a particular anode. Most common

laboratory sources use a Cu anode (Kα=1.54056 Å), although others can be

used. All diffraction data reported in this work is based on monochromatic Cu

69

radiation using a Bruker D2 phase diffractometer (flat plate geometry) in the

School of Chemistry, University of Edinburgh.

The “discovery” of X-rays is often accredited to German physicist Wilhelm

Röntgen in 1895.59 It was not until 1912 that von Laue first theorised that due

to the similar size of X-ray wavelengths and inter-atomic spacings, that X-rays

could scatter from the periodic arrays presented by crystals.60–63 William and

Lawrence Bragg subsequently simplified the models proposed by von Laue

and developed the now famous Bragg’s Law.61,64 This led the Father and Son

to demonstrate the capabilities of X-ray diffraction in 1913 with the structural

solution of NaCl, KCl, KBr and KI,64 and crystallography was born.

Figure 2.4: Schematic representation of Bragg’s Equation. Figure adapted from Ref 58

The model proposed by the Braggs is best represented pictorially, Figure 2.4.

Diffraction is taken to occur from a set of theoretical planes, with interplanar

spacing d, that run through the real space primitive cell. In order to observe

diffraction, scattered X-rays must interfere constructively, and hence must

possess a wavelength, λ, with half integer values of d. For a given λ, this

condition can be met by varying the angle of incidence, θ. This leads to the

Bragg equation,

nλ = 2dsinθ

Equation 2.41

An additional term, n, is observed in Equation 2.41. This term results from the

fact that coherent scattering can occur from higher order reflections in

reciprocal space (wave vectors). It is the convention for this term to be

70

absorbed into d, and to describe sets of planes using the real space Miller

indices, (hkl). These indices describe the number of times a set of planes

intersect with the crystallographic a, b and c axes, respectively. Due to the

intimate relation between d and θ, the positions of the peaks in the diffraction

pattern is indicative of the structure of the unit cell, and hence the

crystallographic parameters that describe the size and shape of the

crystallographic unit cell: a, b, c 𝛼 , 𝛽 and 𝛾 . Coupled to knowledge of the

quantity of electron density located along each plane (by the intensity of

diffraction), X-ray scattering therefore contains the required information to

determine the relative positions of the atoms, and to identify them, within a

crystalline material

2.2.1.1 X-ray Powder Diffraction

If a single crystal scatters monochromatic X-rays, a set of well-defined

diffraction spots are observed according to58

𝐹(ℎ𝑘𝑙) = ∫dV ρ(xyz)exp(2πi(hx + ky + lz)

𝐼(ℎ𝑘𝑙) ∝ |𝐹(ℎ𝑘𝑙)|2

Equation 2.42

Thus, the angle of scattering for a set of Miller indices depends on Braggs law,

Equation 2.41, and their intensities on the relative position of atoms in real

space (electron density, ρ(𝑥𝑦𝑧)), with respect to that set of diffracting planes.

In powder diffraction techniques, however, monochromatic radiation is incident

on a bulk sample, containing many, randomly oriented crystals. This has the

effect of smearing the single diffraction spots into concentric cones, known as

the Debye-Scherrer cones, Figure 2.5.

71

Figure 2.5: Schematic representation of the Debye-Scherrer cones. Adapted from Ref.65

It is common to collect the complete set of cones (e.g. on a 2D area detector),

or simply a small subsection of them (e.g. a point detector). The collected

cones are integrated, and a one-dimensional pattern produced, Figure 2.6. In

accordance with Bragg’s law, Equation 2.41, unit cells of different dimensions

will give rise to a set of peaks at different 2θ positions. Hence, a qualitative

comparison of powder diffraction patterns can be used to assess polymorphic

modifications. In principle, Equation 2.42 suggests that peak intensities

correspond to the relative position of atoms within the diffracting structure.

Peak intensities from powder diffraction experiments can be misleading,

however. If crystallite morphology favours a particular orientation of powder

particles, not all lattice planes will be equally represented within the powder

mixture, a phenomenon known as texturing or preferred orientation. This is

particularly problematic when powder samples are analysed on a flat plate as

is commonly done on the Bruker D2 phaser used in this work. If reliable peak

intensities are required, it is more common to collect diffraction data from

powder samples held within a capillary.

72

Figure 2.6: Example integrated powder diffraction pattern.

2.2.2 Inelastic Neutron Scattering Spectroscopy

Inelastic neutron scattering spectroscopy (INS) is used for studying the

vibrational properties of molecules and materials. An excellent text on its

theory and applications can be found in Reference 66 and 67. In INS

spectroscopy, a beam of neutrons is incident on a sample. These neutrons

scatter from the nuclei within the sample and exchange energy, hence

measurement of vibrational frequencies, (ω) and momentum (𝒒) with the

sample. The INS experiment is often compared to the optical spectroscopies:

Raman and infrared spectroscopy.68 While there are many similarities between

INS and optical probes, the nature of INS possesses several distinct

advantages, including:

1. Ease of modelling.

The scattering function is purely dynamical (Section 2.2.2.2) and as such is

easily calculated within the framework of classical and quantum mechanics.

2. Broad spectral range.

Typical INS spectrometers (e.g. TOSCA) extend from 0 to > 4000 cm-1.69 This

is much broader than typical optical spectrometers, which often miss the far

infrared region 10 − 400 cm-1, and thus omit the lattice, or external, modes of

vibration which are critical in this work.

73

3. Sensitivity to normal modes involving hydrogen atoms.

Optical probes are dominated by heavy atoms due in part to higher electron

densities. INS intensities are proportional to neutron cross sections (𝜎), and

that of hydrogen is particularly high.5

4. Lack of selection rules.

Unlike optical probes, all vibrational modes (including fundamental, overtone

and combination modes) are in principle observed in INS spectroscopy. Group

theory selection rules, which limit the observation of modes in Raman and

infrared spectra, do not apply to INS spectroscopy.

5. Weak interactions with matter.

INS therefore is inherently weighted towards measurement of the bulk

properties of a sample, whereas optical methods are weighted towards surface

properties.

However, these advantages are accompanied by a number of complicating

factors, including:

1. Momentum transfer.

INS does not measure scattering from the centre of the Brillouin zone. While

this does not tend to be a large effect for internal molecular modes, it can lead

to some changes in frequency as compared to optical probes, which only

detect the Brillouin zone centre (i.e. long-range order) vibrations.

2. Neutrons interact weaker with matter than protons.

INS requires much larger sample size, and longer collection times to obtain

vibrational spectra as compared to optical techniques.

5 No theoretical method is available to calculated neutron scattering cross sections 𝜎, and all

tabulated values are directly obtained from experiment. Molecular vibrations are dominated by incoherent neutron scattering, with 𝜎𝑖𝑛𝑐ℎ for common nuclei 1H, 2D, 12C, 13C, 14N and 16O are;

80.27, 2.05, 0, 0.034, 0.5 and 0 barn,83 respectively. 𝜎𝑖𝑛𝑐ℎ of 1H clearly dominates.

74

3. Availability

INS is only possible at dedicated beamlines, located at neutron sources.

4. Temperature Range

Neutron scattering is much more sensitive to temperature than optical probes.

Typical INS spectra are therefore obtained at ca. 10 K.

2.2.2.1 Generation of Neutrons

Neutrons can be produced from a variety of nuclear reactions:67 fusion,

photofission, fission, and spallation. The latter two are most common for

scattering experiments. In a fission reactor (such as the ILL in Grenoble),

neutrons are produced by thermal fission of fissionable isotopes, typically 235U.

Thermal fission of this 235U generates a continuous stream of high energy

neutrons, which can be moderated to produce thermal neutrons. At a spallation

source, high energy protons are generated using a synchrotron. These protons

bombard a metal target (e.g. tungsten or tantalum), which triggers emission of

a cascade of high energy neutrons. As the protons are generated in pulses, so

too are the neutrons at a spallation source. All of the INS spectra used in this

thesis were obtained at a spallation source: the ISIS Neutron and Muon Facility,

STFC Rutherford Appleton Laboratory, UK. Hence this process will be

considered further.

At the ISIS Neutron Facility, high energy protons are generated using a

synchrotron, and are directed at a tantalum coated tungsten target (at Target

Station 1, where the INS beamline is located). These protons have immense

energy (ca. 800 MeV), which excites nuclei in the target. This induces an

intranuclear cascade, which in turn leads to emission of high energy neutrons

(approximately 15 per proton) Hence, an intense neutron pulse is generated.

The neutrons that result from this process have energies on the order of ~ 2

MeV, termed epithermal neutrons. However, these are too energetic for most

practical applications. To reduce the energy of these neutrons, they are

75

therefore passed through a moderator. For the TOSCA beamline, this is

ambient temperature (300 K) water.70 The neutrons undergo numerous

inelastic collisions with the water molecules, and their energy is therefore

subdued. The resulting neutrons (known as thermal neutrons) adhere to a

Maxwell-Boltzmann distribution of energies, about a peak flux that is

characteristic of the moderator. For water, the peak flux is approximately 200

cm-1. It is these thermal neutrons that are finally passed to the instrument and

used for INS.

2.2.2.2 The TOSCA Instrument

The TOSCA instrument was used for collection of INS spectra in this work.69–

71 TOSCA is an indirect geometry time-of-flight (ToF) neutron spectrometer

with resolution Δω/ω ≈ 2 − 3% . As an indirect instrument, the experiment

works by fixing the final energy of the detected neutrons that are scattered

from the sample, and scans the incident energies. As described above, the

incident neutron beam contains a distribution of neutron energies (i.e. it is a

white beam), which are characterised by their kinetic energy and hence the

rate at which they reach the sample. To maximise signal, TOSCA utilises both

forward and backward scattering detectors. Only the neutrons which scatter at

fixed angles (45o or 135o) will impinge on the analyser crystals (the (002) plane

of pyrolytic graphite). It follows from Bragg’s law, Equation 2.41 that since the

scattering Bragg plane is fixed, only a single wavelength (and its higher orders)

of neutron will be passed from the analyser crystal to the detector. All

remaining neutrons will pass through the analyser crystal and are absorbed by

the spectrometer shield. The neutrons that are scattered by the analyser are

passed through a beryllium filter, scattering away neutrons with multiples of

the fundamental wavelength. Finally the remaining neutrons are detected by a

bank of 3He filled detector tubes. The result of using both the graphite analyser

in parallel with the beryllium filters is to create a narrow band-pass filter.

Because neutrons can be treated as both particle and wave, it is possible to

define the kinetic energy of a neutron based on its velocity, v, and its mass,

𝑚𝑛,

76

𝐸 =1

2𝑚𝑛𝒗

2 ⟹ 𝒗 = √(2𝐸

𝑚𝑛)

Equation 2.43

The energy that is transferred between the incident neutron and the sample,

𝐸𝑡𝑟, is defined by the difference in energy of the initial (𝐸𝑖) and final (𝐸𝑓) neutron

energies. For a ToF instrument, the total time, 𝑡𝑡𝑜𝑡, travelled by the neutron is

defined as the time taken to travel from moderator to sample (distance 𝑙1, time

𝑡1), and the time taken for the scattered neutron to travel from the sample to

the detector (distance 𝑙𝑓, time 𝑡2). It therefore follows that

𝑡𝑡𝑜𝑡 =𝑙1𝑣1

+𝑙2𝑣2

=𝑙1

√2𝐸𝑖

𝑚𝑛

+𝑙2

√2𝐸𝑓

𝑚𝑛

Equation 2.44

In Equation 2.44, 𝐸𝑓, 𝑙1 and 𝑙2 are all fixed by the geometry of the instrument.

Hence, the time taken to travel to the detector uniquely defines the incident

energy of the neutron, and correspondingly, 𝐸𝑡𝑟.

2.2.2.3 Neutron Scattering

In a neutron scattering experiment, an incident beam of neutrons (neutral

subatomic particles with mass, 𝑚𝑛, approximately equal to that of a proton) is

scattered from a sample. This scattering can be the result of magnetic

interactions, or due to nuclear interactions. Scattering due to magnetic

interactions is outside the scope of this work, and will not be discussed here;

an introduction can be found in Ref 72.

The total nuclear scattering is defined by the differential scattering cross

section,

77

𝑑σ

𝑑Ω= 𝑏2 =

σ𝑡

4π

Equation 2.45

which describes the amount of total scattering, σt into the elementary

scattering cone of solid angle 𝑑Ω per unit time. This depends on the nuclear

scattering length, 𝑏. The scattering according to Equation 2.45 forms the base

for diffraction experiments.

As was discussed Section 2.2.1.2, INS is an inelastic process, and the

detected neutrons depend both on the solid angle at which they are scattered,

and also on their energy. This is captured in the partial differential cross-

section for a system of 𝑁 atoms, which must therefore be considered as,

𝑑2σ

𝑑Ω𝑑𝐸𝑓= 𝑏2

𝑘𝑓

𝑘𝑖𝑁𝑆(𝑸,ω)

Equation 2.46

where 𝑘 =2𝜋

𝜆 is the wave vector for the incident (𝑘𝑖) and final (𝑘𝑓) neutron.

The term 𝑆(𝑸,ω) is the scattering function, and describes the probability that

the scattering process will change the energy of the system by an amount ω,

and its momentum by ℏ𝑸 = ℏΔ𝒌. Because both the energy and angle of the

final scattered neutron are fixed on TOSCA, this also fixes 𝒌𝑓. The value of 𝑸

is therefore dependent on 𝐸𝑡𝑟, and TOSCA probes a narrow stripe in (Q,ω) of

kinematic space.70

The scattering described in Equations 2.45 and 2.46 account for the elastic

and inelastic scattering processes. However, due to nuclear effects (isotope

and spin effects), the scattering cross section in both cases requires

consideration of coherent and incoherent processes. The former describes the

in-phase scattering and results in interference effects. Hence, coherent

scattering is required for diffraction and vibrational spectroscopy of collective

motions (e.g. phonon dispersion). The incoherent scattering describes motion

78

of single particles, in which no correlation exists between different molecules

or atoms. These motions are particularly important for studying internal

molecular vibrations. In an INS spectrum, both coherent and incoherent

scattering processes are observed, and the dominating scattering mechanism

depends largely on the scattering cross-sections of the atoms involved.

Incoherent scattering dominates in hydrogen-containing compounds.

The absolute intensity of an INS spectrum is difficult to interpret, and so only

the relative spectral intensities are considered. The calculated relative intensity

(the scaled scattering factor, 𝑆∗(𝑸,ω𝑖) ) of the 𝑖𝑡ℎ vibrational mode with

momentum transfer 𝑸 and neutron energy loss 𝐸𝑡𝑟 = ω𝑖 is defined as73

𝑆∗(𝑸,ω𝑖)𝑚𝑛 = 𝑦σ𝑚

[(𝑸 ∙ 𝑖𝒖𝑚)2]𝑛

𝑛!𝑒𝑥𝑝(−(𝑸 ∙ ∑ 𝑢

𝑖𝑚

𝑖

)

2

)

Equation 2.47

In Equation 2.47, 𝑖𝒖𝑚 is the displacement vector for atom m of mode i, y is a

linear factor (units barn cm), which acts to convert the actual units of 𝑆∗(𝑸,ω𝑖))

into scaled dimensionless units. The final variable, n, indicates the final state

of the excited mode. So, an elastic process has n=0, a fundamental n=1, the

first overtone n=2, etc. The pre-exponential term of Equation 2.47 increases

with increasing momentum transfer and vibrational amplitude. The exponential

term (known as the Debye-Waller factor), however, decreases more rapidly

with 𝑸2𝒖2. Hence, there is an overall decrease in vibrational amplitude with

increasing temperature. As such, it is typical to collect INS spectra at cryogenic

temperatures, although (as done in this thesis) higher temperature

measurements are still possible.

Equation 2.47 is surprisingly simple, and depends on the momentum

transferred, the scattering cross section and the magnitude of the atomic

displacement. Hence, the observed INS intensity is purely dynamic and can

be easily calculated: frequencies are obtained from normal mode eigenvalues

79

and the displacements from the eigenvectors. Thus INS is an excellent

technique against which to validate calculated vibrational spectra.31,74.

2.2.3 BAM fall Hammer

A number of tests have been developed to measure the impact sensitivity of

energetic materials (EMs). These include the Picatinnany Arsenal apparatus,

the Bureau of Mines Machine, the Rotter Impact Machine, and the BAM fall

hammer.75,76 The standard procedures and device depend largely on

geography.77 Some devices, such as the Rotter Impact Machine, define a

successful initiation based on the production of gas products. For other

methods, such as in the BAM fall hammer, a successful initiation is determined

by the user, and is generally based on sound or visual inspection of the sample.

It follows that the reported impact sensitivities that result from different testing

methods can vary substantially.78 Further difficulties in the experimental

validation of impact sensitivity stems from sample and environmental factors,79

which include particle size, crystallinity, purity, temperature, and humidity,

amongst others. These factors are not regularly controlled, and their effects on

impact sensitivity are poorly understood. These all contribute to the extreme

variability in reported impact sensitivities for materials.

80

Figure 2.7: BAM fall hammer device used for impact sensitivity testing. (A) The BAM BFH-12

apparatus. (B) Sample anvil. Figure adapted from Ref. 80

This work makes use of the BAM fall hammer (BFH-12), based at the

Cavendish Laboratories, University of Cambridge, Figure 2.7A. This device

has been accepted as the NATO qualification testing method, described in the

United Nations recommendations for the Transport of Dangerous Goods.76

The BFH procedure requires a sample of 40 mm3 to be enclosed in an anvil

setup, composed of two coaxial steel cylinders and a guide ring, Figure 2.7B.

To apply the impact, a load mass of 10, 5, 2, 1 or 0.5 kg is dropped onto a

sample, from heights ranging 20 – 100 cm. The energy is therefore calculated

as, e.g. 1 kg from 50 cm = 5 J.

81

Figure 2.8: Probability of initiation of energetic materials to impact. Figure from Ref. 81

There are two testing protocols typically used for assessing the impact

sensitivity of EMs. The Limiting Impact Energy Test (known also as the 1-in-6

method) is outlined in the United Nations testing requirements for the transport

of explosives.76 According to this method, the impact energy is lowered step-

wise until an impact energy is reached at which six consecutive trials result in

a ‘No Go’ (that is, no explosion, discolouration, flash or loud noise is observed

upon impact). This is taken as the highest drop height at which impact will not

induce initiation of the sample, ℎ0 , Figure 2.8. This method is good for

assessing the safe handling of materials, and for testing small sample sizes.

The Bruceton up-down method also leads to measurement of a sample’s ℎ50

value, Figure 2.8 (i.e. the drop height at which 50% of tested samples will

initiate). It has a corresponding energy denoted 𝐸50. In this method, the drop

height is varied depending on the outcome of a previous measurement. If the

sample initiates at a particular energy, the subsequent test is performed at a

lower stimulus energy. Instead if the sample does not initiate, the new test is

performed at higher energy. This method assumes a normal response to input

energy and has been criticised,82 although it remains widely used.

82

Both methods are indicative of the impact sensitivity of a material. However,

their values are clearly not directly comparable. Moreover, the fall hammer

method used is often not described in the literature alongside the numerical

data, and this further contributes to the large discrepancy of sensitivity reported

in literature. Where possible, this work seeks to compare its models against

reported ℎ50 values, unless otherwise stated.

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Chapter 3

VIBRATIONAL UP-PUMPING: PREDICTING IMPACT

SENSITIVITY OF SOME ENERGETIC AZIDES This chapter published as Michalchuk et al (2018), J. Phys. Chem. C. 122 (34) 19395-19408

and Michalchuk et al (2018), Phys. Chem. Chem. Phys., 20, 29061-29069

3.1 Introduction

To a simple approximation, a mechanical impact can be taken to induce two

main effects: (1) the material being impacted is compressed, and (2) if the

impact energy exceeds a threshold energy, fracture of the impacted body.1 In

the first, a compressive pressure wave passes through the material, akin to an

acoustic wave. The propagation of this pressure wave through the material has

been suggested to induce vibrational excitation of the lattice by a two-fold

mechanism.2 The pressure associated with the impact leads to a shift in the

vibrational frequencies of the material. These vibrations are subsequently

populated by the sudden increase in energy of the lattice. For the adiabatic

compression of a solid,3

(𝑇

𝑇𝑜) = (

𝑉

𝑉𝑜)−𝛤

Equation 3.1

where Г is the Grüneisen parameter describing the vibrational anharmonicity

of the lattice, V/V0 describes the change in volume on compression, and T/T0

is the change in temperature of the bulk material on compression. The exact

temperature that can be achieved depends largely on the heat capacity of the

material.4 It follows that the magnitude of the excitation that results from an

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impact is proportional to the magnitude of the impact pressure, and the

response of the material to this pressure.5–7

Previous work found that for the model organic material naphthalene, a modest

impact (4 GPa) is associated with a total increase in internal energy, ΔU, of 37

000 cm-1 per molecule.4 However, as ΔU = H + W, the increase in internal

energy will distribute between heat (𝐻) and work (𝑊). The proportion of ΔU

that converts to 𝐻 follows from Equation 3.1, and therefore increase with the

anharmonicity of the material through Γ. For naphthalene, V/Vo = 0.793 at 4

GPa, while for α-NaN3 this value is ca. 0.860,8 and ca. 0.915 for AgN3.9 Thus

the values of Γ(NaN3)10 ≈ Γ(AgN3)11, whilst Γ(naphthalene) is nearly 2.5-fold

higher.4 The proportion of impact energy that is converted to heat is therefore

expected to be lower for the inorganic azide materials. The energy that is

introduced into the material affects only the lattice vibrations, which equilibrate

very rapidly. This leads to a highly excited phonon region, or bath.5,6,12 Noting

that the heat capacity of the phonon bath is much lower than for the bulk

material, it was found that the phonon quasi-temperatures reach approximately

2300 K, corresponding to ca. 1.2 eV per molecule in napthalene.4

In addition to the impact-induced heating of the material, a second process,

fracture, can occur.1 Within the contact surface, stresses build beyond the

elastic limit of the material. This leads to plastic deformation in the form of

dislocations or fracture.13 Analogous to the rupture of a loaded spring, the

sudden rupture of non-covalent (or covalent) interactions along a fracture

surface leads to rapid excitation of lattice vibrations associated with the

ruptured interaction.14 Rapid equilibration again leads to formation of a ‘hot’

phonon bath.

The hot phonon bath produced by both mechanisms is bound by a maximum

frequency, defined as Ωmax, Figure 3.1.15–17 Vibrational modes that sit above

the phonon bath are not directly excited by the impact, and remain vibrationally

‘cold’. The excess vibrational energy of the phonon bath can either dissipate

outwards, or upwards; the latter is substantially faster and hence vibrational

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energy continues to scatter upwards. The general process of up-conversion

occurs in two stages:15

1. Scattering of the phonon bath modes leads to excitation of intermediate

librational modes, known as doorway modes.

2. The ‘hot’ doorway mode subsequently scatters with additional phonon

bath modes.

This process ultimately allows up-conversion of energy from the initially excited

phonon region into the localised molecular modes, Figure 3.1. Experiment has

found such energy conversion processes to occur on the order of

picoseconds,16 and occur more rapidly around defects.4,18 These processes

therefore occur considerably faster than the thermally-induced chemical

decomposition in energetic materials (EM).12 The rate of vibrational energy

transfer is on the same time scale as the primary events associated with an

explosion,6 and has prompted interest in this model to explain mechanically-

induced initiation of EMs.19,20

Figure 3.1: Schematic representation of the vibrational energy ladder traversed by mechanical

(shock) impact energy. Injected energy begins in the delocalised phonon bath, up-converting to

the localised molecular-based target modes via intermediate doorway modes. Figure adapted

from Ref. 21.

90

The initial stage of a chemical explosion involves the rupture of a covalent

bond. Within the proposed vibrational energy transfer model,21,22 it follows that

the vibrational energy must ultimately localise into a particular molecular

vibration, the target mode, 𝑄𝑇. When this target mode is sufficiently populated,

distortion along its eigenvector reduces the energy separation of the frontier

orbitals, and athermal bond rupture ensues.23–25

The apparent structural simplicity of azide-based EMs, where only a single

covalent bond (N-N bond) exists for initiation, offer a particularly interesting

challenge for understanding impact sensitivity. Azides have been known for

over a century,27 and their initiation mechanism has been the subject of

research for nearly three-quarters of a century.28 Despite this apparent

simplicity, the sensitivity of these materials span orders of magnitude. For

example, NaN3 is completely insensitive to impact, while Pb(N3)2 is a common

primary explosive used in detonators.13 Broadly, the azide materials can be

classified into three structural types: ionic, polymeric and molecular.29 The

ionic materials have been reported to be less sensitive than the polymeric or

molecular materials.30 The existence of covalent bonds in the latter two

material types has been used to rationalise a decrease in N-N bond strengths

and hence increase sensitivity, although the change in N-N dissociation

energies typically varies only slightly.29 This rationale is therefore insufficient

to explain the range of sensitivities observed, both within and between

structural classifications, or between polymorphic forms.31

The breadth of physical and chemical characteristics displayed by azide-based

EMs is vast, which in conjunction with the relative simplicity of azide-based

chemistry,32 continues to keep the development of azide-based materials an

active area of research.32–35 However, without a means to predict the relative

sensitivity of new materials a priori, the preparation of new azide materials

remains very hazardous.

91

3.2 Aims

Simple inorganic azide energetic materials display a huge variation in impact

sensitivity behaviour, and development of new azide-based materials is of high

interest to the energetics community. A physical basis for their initiation has

not yet been elucidated, although some qualitative trends, including bond

lengths, cation ionization potentials and symmetry breaking of azide vibrational

modes with energetic behaviour have been noted.36,37 More physical models

have also been proposed, noting that crack propagation rates may be linked

to sensitivity,13 based on formation of hot-spots at crack tips or deformation

pile-ups. However, no unified mechanism has yet been proposed to explain

the sensitivity relationships observed for these compounds. The sensitivity of

energetic materials is a complex phenomenon, and the underlying mechanism

can differ. For example, initiation may occur due to hot-spot formation in a bulk

composition, or at extended defects within single crystallites (Chapter 1.2).

Which mechanism dominates (and thus the type and magnitude of hot-spot

that forms) depends on the nature of the prepared sample and is largely

irreproducible. Regardless of the mechanism by which the energy is generated,

its localisation can be sought in terms of the vibrational up-pumping

mechanism (Chapter 1.2.2). At the most fundamental level is the intrinsic

sensitivity of a material. This describes the propensity of the ideal material (i.e.

defect-free) to react under mechanical perturbation and will form the basis for

the work presented here.

This chapter aims to build a model for ideal crystalline materials, based on a

vibrational up-pumping approach, in order to rationalise and predict the relative

sensitivity ordering of a test set of azide-based EMs. To that end, the work

presented here sought to:

1. Identify a vibrational mode (target mode, 𝑸𝑻) that is responsible for the

initial decomposition of the explosophoric azido anion.

2. Investigate the pathways to vibrational up-pumping.

3. Correlate the relative rate of vibrational up-pumping to impact sensitivity.

92

3.3 Test Set of Energetic Azides

The model presented here is constructed from a selected series of crystalline

energetic azides, Table 3.1 and Figure 3.2, selected to cover a range of

reported experimental sensitivities and cover the three main structural types.

Of the ionic species selected, two are based on molecular cations:

triaminoguananidinium azide (TAGZ) and ammonium azide (NH4N3). The

experimental measurement of EM sensitivity is highly unreliable, with many

conflicting reports in the literature.38 In many cases, conflicting reports are due

to crystal size, purity, defect concentration, as well as both environmental and

experimental conditions.39–41

Literature discrepancies are particularly prevalent across the azide materials,

most notably for the ordering of the more sensitive materials. Due to these

large discrepancies, exact values are not quoted in developing the model in

this chapter, but instead the relative ordering is considered. The sensitivity

classifications given in Table 3.1 are based on the following:

1. Ba(N3)2, is quoted by some as being more sensitive than AgN3,36 and

by others as less sensitive.36,42 Its impact sensitivity has been measured

to be between 4-10 J and shown to be highly dependent on temperature,

particle size and impurities.30,43,44

2. AgN3 is generally accepted to be somewhat less sensitive than

Pb(N3)2.44

3. Zn(N3)2 in its pure form remains poorly characterized. Unquantified

reports indicate that it explodes on minimal mechanical provocation,34

with mixtures of zinc with Pb(N3)2 forming dangerously sensitive azide

products.45 Literature reports that state “zinc azide” to be insensitive are

likely inadvertently discussing the sensitivity of hydrated form, given the

extreme hygroscopicity of the anhydrous form.46,47,30

4. LiN3 has been measured to have an impact sensitivity of ca. 22 J,30 with

some sources stating it to be completely insensitive.36

93

5. Sn(N3)2 is believed to have a similar sensitivity as its structural

homologue, Pb(N3)2. 33 (ca. 1.7 J 44).

6. NaN3 and NH4N3 are known to exhibit very low sensitivity.48 However,

recent work has suggested that impacts of ca. 25 J can induce chemical

decomposition in NaN3 without the visible burn that is typically used as

the main criterion for experimental impact sensitivity testing.49

7. TAGZ has been found to have impact sensitivity of ca. 34 J.50

8. Both liquid and gaseous HN3 are known to be highly sensitive,30

although no sensitivity studies on crystalline51 HN3 are known.

The work presented in this chapter therefore compare against a general

experimental ordering of NaN3 ≈ TAGZ ≈ NH4N3 < LiN3 < Ba(N3)2< AgN3 <

Sn(N3)2, with the exact positions of HN3 and Zn(N3)2 remaining unknown.

Table 3.1: Test set of energetic azides used in this work, listed in approximate order of increasing

sensitivity.

Material Sensitivity Class* Bond Type# Ref.

NaN3 I I 42

TAGZ LS I 50

NH4N3 LS I 48

LiN3 LS I 30

HN3 S M 30

Ba(N3)2 S P 30,42,43

AgN3 S P 31

Zn(N3)2 S P 34

Sn(N3)2 S P 33

*Experimental sensitivity class according to indicated references. Sensitivity reported as

insensitive (I), low sensitivity (LS), sensitive (S). #Classification of bonding type: molecular (M), ionic

(I) or polymeric (P)

94

Figure 3.2: Conventional crystallographic cells of the energetic azides used in this work. The space

group (SG) is given for each cell, along with an indication of the crystallographic axes. The azides

are given in approximate order of impact sensitivity according to literature reports. In all cases,

atoms are coloured as: blue- N; grey- metal; white- hydrogen; black- carbon. The structure of the

triaminoguanidinium (TAG) cation is shown above its unit cell.

3.4 Methods

Gas phase calculations. Calculations of isolated molecules were performed in

vacuo using Molpro 2012.52 Geometry optimisation and subsequent vibrational

frequency calculation was performed to ensure equilibrium geometry was

obtained. The electronic structure was built upon a CAS(8,8)/6-31+G*

calculation, on which a MRCI calculation was added for the same active space

95

and basis set. The highest atomic orbital was chosen to ensure inclusion of

frontier bonding orbitals, while also ensuring that the orbital occupation of the

highest unoccupied orbital was no more than 0.02.53

Condensed Matter Calculations: Input unit cell geometries were taken from the

experimentally determined structures deposited in the Inorganic Crystal

Structural Database (ICSD, FIZ Karlsruhe), Table 3.2. Geometry optimisation

was performed using plane wave Density Functional Theory (PW-DFT) as

implemented in CASTEP v16.54 Where appropriate pseudopotentials were not

available in CASTEP, calculations were performed using the

QuantumEspresso v6.1 software. For CASTEP calculations, the GGA

functional of Perdew-Burke-Ernzerhof (PBE)55 was applied, while

QuantumEspresso calculations used the rVV10 non-local functional, which

performs very similarly to the PBE+D scheme.56 For CASTEP calculations,

dispersion correction leading to the best structural agreement with experiment

was chosen: Grimme’s D257 dispersion correction, PBE-D2, or that of

Tkatchenko-Scheffler (TS), PBE-TS.58 The use of PBE + dispersion has

previously performed well for these materials.33,59 Convergence criteria for

electronic structure calculations are given in Table 3.2. In all cases, the

electronic structure was sampled on a Monkhorst-Pack k-point grid60 with

spacing no greater than 0.05 Å-1. Note that tighter convergence was required

for the Zn structure to remove imaginary phonon frequencies. Norm-

conserving pseudopotentials were used throughout. The optimised structural

parameters are given in Table 3.3.

Phonon calculations were performed on the optimised structures, using the

same computational packages as for structural optimisation, within the

framework of linear response theory. Dynamical matrices were initially

calculated on a regular grid of q-points and subsequently Fourier interpolated

onto a finer grid. Phonon dispersion curves were generated along the high

symmetry paths as suggested by SeeKPath61.The dynamical matrix was

subsequently calculated across a regular set of q-points. Density of states

(DoS, 𝑔(ω) ) were generated using Gaussian line broadening of 10 cm-1.

96

Phonon density of states were normalised to 3N, where N is the number of

atoms, such that the resulting curves represent the number of available

coupling pathways within each unit cell. Note that phonon dispersion curves

were generated for the primitive cells in all cases except for AgN3, for which

imaginary frequencies could only be removed by use of the conventional cell.

This does not affect the structure of 𝑔(ω), but only the factor of 3𝑁, which can

be accounted for by re-normalisation.

Band structures were generated along high symmetry paths. As GGA based

band gaps are known to provide poor agreement with experiment, further band

structures were generated in CRYSTAL1763 using the HSE0664 hybrid DFT

functional (with localised basis sets available from the CRYSTAL17 database:

N- N_m-6-311G(d)_Heyd_200565; Na, Li, C; H-TZVP65; Ba-HAYWSC-

3111(2d)G 66; Ag-From Ref 67; and Sn- DURAND-21G*68) which has been

demonstrated to offer reasonable agreement with experimental band gaps for

a broad range of materials.69 Those presented here can therefore be regarded

as accurate to within a reasonable level of confidence. The wavefunction was

converged to < 10-8, and convergence criteria TOLINTEG 7 7 7 19 30, as

recommended for this functional and basis set65. The electronic structure was

calculated at 172 k-points across a 7 x 7 x 7 Monkhorst-Pack grid.60 Analysis

of the crystal overlap Hamilton populations (COHP) were performed using the

properties code, as implemented in the CRYSTAL17 suite. A minimal basis set

(STO-6G) was used to avoid spurious overlap of basis functions, which were

found to contaminate the calculation. COHP were calculated for directly

bonded N atoms in the azido anion.

Inelastic Neutron Scattering Spectroscopy. All spectra were collected on the

TOSCA spectrometer at the ISIS Neutron and Muon source.94-96 A sample of

NaN3 (ca. 1.5) was placed in an aluminium sample holder and cooled to ca. 10

K. Data were collected for a total of ca. 400 μAh. Both forward and back-

scattered data were summed and corrected for scattering from the sample

holder and background. All data processing was done using Mantid.72

Simulated spectrum was generated using ABINS,97 as implemented in Mantid.

97

Table 3.2: Optimisation criteria for the energetic azides. The quantum chemical code (QuantumEspresso; Q, or Castep; C), is shown alongside the applied

exchange correlation scheme (XC).

Azide ICSD Code

Code XC 𝚫𝑬 /eV.atom-1

Max. Force / eV Å -1

Max. Atomic Disp. /Å

Max. Stress/ GPa

𝑬𝒄𝒖𝒕 /eV

NaN3 29370 C PBE + D2 2x10-6 0.001 0.001 0.005 1800

NH4N3 2236 C PBE +TS 2x10-6 0.001 0.001 0.005 1800

TAGZ Ref 62 C PBE + TS 2x10-6 0.001 0.001 0.005 1800

HN3 261955 C PBE + TS 2x10-6 0.001 0.001 0.005 1800

LiN3 34675 Q rVV10 1x10-9 1.0 x 10-8 Ry/Bohr 0.0001 5x10-6 2312

Zn(N3)2 430428 C PBE+D2 2x10-9 0.0005 0.0005 0.0005 1800

Ba(N3)2 26202 Q rVV10 1x10-9 1.0 x 10-8 Ry/Bohr 0.0001 5x10-6 2312

AgN3 88335 Q rVV10 1x10-9 1.0 x 10-8 Ry/Bohr 0.0001 5x10-6 1768

Sn(N3)2 433812 C PBE + TS 2x10-6 0.001 0.001 0.005 1800

98

Table 3.3: Comparison of experimental (exp) and computed (calc) unit cell geometries. Low temperature experimental data are used where available

Azide a b c α β γ V 𝚫V /%

AgN3 (Exp) 5.60 5.98 5.99 90.00 90.00 90.00 200.86 +1.5

AgN3 (Calc) 5.71 5.96 5.98 90.00 90.00 90.00 203.81

BaN3 (Exp) 9.59 4.39 5.42 90.00 99.75 90.00 224.89 +6

BaN3 (Calc) 9.83 4.44 5.53 90.00 99.14 90.00 238.50

HN3 (Exp) 8.21 8.21 6.78 110.42 110.42 90.01 397.42 +5.9

HN3 (Calc) 8.38 8.38 6.90 110.51 110.51 90.00 421.18

NH4N3 (Exp) 8.93 3.81 8.66 90.00 90.00 90.00 294.62 +0.1

NH4N3 (Calc) 9.02 3.81 8.57 90.00 90.00 90.00 294.80

LiN3 (Exp) 5.63 3.32 4.98 90.00 107.40 90.00 88.73 -0.5

LiN3 (Calc) 5.59 3.32 4.91 90.00 104.80 90.00 88.25

NaN3 (Exp) 3.61 3.61 5.41 105.36 105.36 60.96 57.30 +1.1

NaN3 (Calc) 3.59 3.59 5.20 101.81 101.82 61.76 57.94

SnN3 (Exp) 6.78 11.06 6.23 90.00 94.67 90.00 465.51 +0.7

SnN3 (Calc) 6.69 11.80 5.95 90.00 91.41 90.00 468.93

TAGZ (Exp) 6.68 7.72 13.14 90.00 95.44 13.14 674.80 +0.5

TAGZ (Calc) 6.69 7.74 13.16 90.00 95.76 90.00 678.39

ZnN3 (Exp) 3.46 16.26 6.93 90.0 95.90 90.0 387.80 +1.5 ZnN3 (Calc) 3.44 16.47 6.99 90.0 96.42 90.0 393.84

99

3.5 Results and Discussion

3.5.1 Bond Rupture of Explosophoric 𝐍𝟑−

The initiation of an energetic material involves rapid release of chemical

potential energy. This process must therefore involve rupture of a covalent

bond within the explosophoric moiety of the material. Within the azide

materials, this is rupture of an N-N bond. In their ground state structures, the

azide materials contain a closed-shell N3− molecule. It is therefore necessary

to understand the reactivity of this molecule. Due to the delocalised nature of

electronic states in solids, however, there is an intimate interaction between

the electronic states of the counter-ion and the azido anion. Further, with the

unavoidable presence of intrinsic defects (e.g. vacancies), the electronic band

structures will likely include some additional states within the band gap that

may influence sensitivity.70,71 A variety of pathways are therefore available for

the reduction or oxidation of the azido anion species within the solid state. Only

those in the ideal crystal are considered through this chapter.

Figure 3.3: Electronic structure of the azido anion by HSE06 in a periodic box, N3-. (A) Projected

Crystal Overlap Hamilton population (pCOHP) for a TZVP (blue) and STO-6G (black) basis set. A

high-level basis set leads to spurious results,72 hence use of a minimal basis set (B) The ‘density

of states’ for the azido anion, alongside visualisation of the associated molecular orbitals. Figure

from Ref. 21

The electronic structure of the azido anion, present in all the energetic azides

studied here, is given in Figure 3.3. The Projected Crystal Overlap Hamilton

100

Population (pCOHP) weights each orbital by the overlap matrix component

associated with neighbouring N atoms. Hence, it identifies orbitals in which

directly bonded N atoms are stabilised or not. The first excited state associated

with the azido anion involves excitation of an electron from a non-bonding πg

orbital (i.e. –pCOHP close to zero) to an antibonding πu molecular orbital (i.e.

–pCOHP is negative). Hence excitation should yield considerable weakening

of the N-N covalent bonds. Analysis of the crystalline band structures in Figure

3.4 suggests that for all azides, the top of the conduction band and bottom of

the valence bands are both primarily azide in character. This suggests that

direct excitation of the N3− molecule is the dominant transition mechanism.

This concept has led to the ‘band gap criterion’ for predicting impact sensitivity

in some materials Chapter 1.3.2.2).73 Despite interest in the energetic azides,

no data on the experimental band gaps could be found. As such, band gaps

presented here are compared against literature calculated values, where

possible, Table 3.4. The electronic band structure was calculated for each of

the nine crystalline azide materials listed in Figure 3.1 using the PBE and

HSE06 (Shown in Figure 3.4) DFT functionals. It is first worth noting that all of

the crystalline materials have smaller band gaps than the isolated molecule.74

This is expected and due to the periodicity of the former. It is generally found

that the ionic azide materials (NaN3, TAGZ, NH4N3, LiN3 and BaN6) exhibit

larger band gaps than the polymeric (SnN6, AgN3, ZnN6) or molecular (HN3)

systems. Noting that the ionic azide materials are typically less sensitive to

mechanically-induced initiation than the polymeric and molecular systems

(Section 3.3), the trend in PBE band gaps generally agree with the band gap

criterion. However, it is worth noting a discrepancy in this trend, where the

sensitivity of BaN6 >> NaN3 despite their band gaps being similar.

As expected, the magnitude of the band gap for each of the materials

increases on moving to the screened hybrid HSE06 functional, Table 3.4.

Comparison to previously calculated band gaps, based on various functionals,

for some of the azide materials agree well with the HSE06 values. A notable

exception is NH4N3, for which earlier works have suggested a direct band gap

101

of ca. 0.7 eV lower than that calculated here.75 Other theoretical studies

support the present findings that NH4N3 is an indirect band gap material.76

Experimental validation is therefore required. The band gap of Ba(N3)2 is also

considerably larger by HSE06 than previously reported based on PW91 GGA

DFT calculations. Despite this GGA functional performing relatively well for

other azides, it appears to fail in the case of Ba(N3)2. Instead, the PW91 result

for Ba(N3)2 is much closer to the PBE band gap.

Table 3.4: Calculated band gaps for the crystalline azide materials by PBE (PBE and LBS) and HSE06

(LBS). The band gaps are labelled as being direct (D) or indirect (ID). Comparison of energies are

shown with literature and discrepancy in momentum conservation is shown.

Material PBE HSE06 D/ID Literature

N3− 5.64 7.06 -- --

α-NaN3 4.02 5.34 D 5.38҂a ; 5.03ⱽb

TAGZ 4.48 5.82 ID --

NH4N3 4.36 5.75 ID 5.08 (D) ҂b

LiN3 3.56 4.75 D 4.98҂a ; 4.68ⱽb

HN3 3.78 5.20 ID --

Ba(N3)2 4.12 5.32 D 3.65ⱽc

AgN3 1.57 2.77 D 1.72٣a ; 2.5ⱽa

Zn(N3)2 3.41 4.78 ID --

Sn(N3)2 0.66 1.51 ID --

҂ TB-mBJ band gaps from (a) Ref. 59 (b) Ref. 75 ; ٣ PBE band gap from (a) Ref 77 ; ⱽPW91 band gap

from (a) Ref 78 (b) Ref 79 (c) Ref 80

Interestingly, while the band gap criterion holds relatively well across the PBE-

based band gaps, it is less prominent for the higher-level functional HSE06.

Noting that the latter is expected to be much more accurate, this suggests that

the correlation with the lower level functional was largely fortuitous. Some

earlier works also show discrepancy between sensitivity and band gap.81 No

obvious trend is seen between the sensitivity and a material having a direct or

indirect electronic band gap.

102

Figure 3.4: Electronic band structures (HSE06) for the energetic azides. Band dispersions are plotted along high symmetry lines. The partial density of states

are plotted for each, decomposed as (blue) N3− channel and (green) cation channel.

103

3.5.1.1 Dissociation of 𝑵𝟑−

Throughout the following discussion, the symmetric N-N bonds are denoted 𝑅1

and 𝑅2, with the electronic state given as a preceding superscript. The 𝑆𝑜 bond

length of 𝑅1 is thus denoted 𝑆𝑜𝑅1.

In the closed-shell ground state of the N3− anion, the optimised geometry

(CAS(8,8)/6-31+G*) of the molecule shows two equivalent N-N bonds ( R1 S0 =

R2 S0 = 1.1865 Å), with the bond angle θ𝑁𝑁𝑁 = 180o. This value is in very good

agreement with the experimentally-observed gas-phase equilibrium bond

length of 1.188 Å, as obtained from rotational spectroscopy.82 If the

optimisation is performed with a full valence active space, CAS(16,12), the N-

N bonds elongate, with R1 S0 = R2

S0 = 1.2002 Å. This is only slightly longer

than the experimental bond length in the gas phase molecule, and for the

molecule in the ionic azide materials (ca. 1.17-1.18 Å). Optimisation of the first

triplet state, 𝑇1, leads to an increase in the N-N bond lengths, with R1 T1 = R2

T1 =

1.2607 Å using a CAS(8,8) active space, increasing to 1.2855 Å when

optimised at CAS(16,12). The 𝑇1 state remains linear in all cases. Similarly,

the bond lengths of the first singlet state, 𝑆1, expand from 1.2243 Å to 1.2475

Å when moving from a CAS(8,8) to CAS(16,12) calculation.

Importantly, when the active space is increased, the relative energies of the

equilibrium structures change only slightly with E𝑆1 − 𝐸𝑆0 < -0.3 eV, E𝑇1 − 𝐸𝑆0,

< +0.03 eV. The same holds for comparison of structures with R2 = 2.5 Å, i.e.

beyond the dissociation limit of the azido anion. The effect of increasing the

active space is therefore small relative to the additional computational costs,

and as such the smaller active space was used for the remainder of this work.

To assess the effect of vibrational normal coordinates on the relative stabilities

of the electronic states of N3−, all excitations were performed as Frank-Condon

(FC) transitions from perturbations to the ground state (𝑆0) optimised geometry.

Hence, rupture of bond 𝑅2 along the excited state PESs are investigated

based on 𝑅1 fixed at the optimised length of the 𝑆1 state.

104

Elongation of 𝑅2 leads to bond dissociation at R2 S0 > 2 Å, with a dissociation

energy of ca. 4.52 eV, Figure 3.5A. The |𝑆0, 𝑉0⟩ |𝑆1, 𝑉0⟩ transition requires ca.

5.1 eV energy, with the FC transition requiring 5.22 eV. In contrast to the

dissociation energy of the ground state, N-N dissociation in the S1 state has an

energy barrier of only ca. 1 eV. This occurs with R2 S1 > 1.75 Å. Importantly,

once this energy barrier is surpassed, dissociation is spontaneous, with

ΔE( R2,diss S1 − R2,eqm

S1 ) ≈ −0.4 eV . The |𝑆0, 𝑉0⟩|𝑇1, 𝑉0⟩ transition occurs at

notably lower energy, 4.21 eV, with the FC transition occurring at 4.46 eV. In

the T1 state, bond dissociation is met with a similar energy barrier to the S1

state (ca. 1 eV when R2T1 > 1.65 Å). Again, once this energetic barrier has been

surpassed, bond dissociation is spontaneous, with ΔE(T1R2,diss - T1R2,eqm) ≈ -

1.05 eV. The dissociation product of the T1 state sits ca. 1.2 eV below that of

the S0 and S1 states. The same general trend is observed for all higher excited

states. It therefore follows that excitation of the azido anion into any of the

excited states favours bond dissociation.

Based on energetic considerations, the T1 state appears the most likely

candidate for bond dissociation given its low dissociation barrier. However, at

equilibrium geometry, the energies required to reach any of the excited states

greatly exceeds kBT. As described in Section 3.1, impact induced initiation

results from mechanical perturbation of the impacted material. This leads to

excitation of the lattice modes, and eventual localisation of this energy into

vibrational modes.4 The amount of energy localised in this way can be greatly

in excess of the energy achievable by bulk temperatures, and sufficient to

induce bond rupture.12 Hence, it is necessary to consider the effects of the

vibrational normal coordinates on the electronic structure of N3− . For the

isolated anion, these include two degenerate bending modes (δθNNN), a single

symmetric stretch (δRS), and an asymmetric stretching mode (δRA). Variations

in the electronic structure of N3− were studied as a function of these normal

modes, Figure 3.5B-D.

105

Figure 3.5 Potential energy surfaces (PES) associated with the N3

− anion. PES are shown for (A)

elongation of a single N…N covalent bond, and the three symmetry independent normal modes:

(B) δθNNN, (C) δRS, and (D) δRA; r1=(reqm + α/10), r2=(reqm - α/10), where reqm is the equilibrium

bond distance. In each case the potential energy surface for S0 (black), S1 (red), S2 (blue), T1

(pink) and T2 (green) are given. All energies are normalized to the S0 equilibrium energies. Figure

adapted from Ref. 21

At the lowest frequency, δθNNN is most responsive to mechanical perturbation.

As the anion bending angle θNNN deviates from 180𝑜, the energy of the ground

state species increases until an apparent plateau is achieved at ca. 110𝑜 ,

Figure 3.5B. At this plateau, an overall increase in the internal energy, ΔU, of

3.9 eV is observed. In contrast, as θNNN decreases, the energy of the S1 state

106

decreases, reaching its minimum energy at approximately 140𝑜 , with

𝑆1𝐸(θ𝑁𝑁𝑁 = 180𝑜) − 𝑆1𝐸(θ𝑁𝑁𝑁 = 140𝑜) ≈ 1.1eV. At this angle, the energy

separation between the S0 and S1 state decreases from ca. 5.3 eV to only ca.

3.0 eV. This energy gap decreases further as θNNN continues to decrease,

reaching a minimum energy separation of 1.5 eV at 115𝑜.

The energy of 𝑇1 also decreases with 𝜃𝑁𝑁𝑁. An energetic minimum is observed

at ca. 𝜃𝑁𝑁𝑁 = 1300, where 𝑇1𝐸(θ𝑁𝑁𝑁 = 180𝑜) − 𝑇1𝐸(θ𝑁𝑁𝑁 = 130𝑜) ≈ 1.7 eV.

At this angle, the energy separation between 𝑆0 and 𝑇1 reduces from 4.2 eV to

only 0.7 eV. This energy is less than the energy associated with the second

overtone of 𝛿𝑅𝐴. As 𝜃𝑁𝑁𝑁 is compressed further, a conical intersection (CI) is

reached, with an 𝑆0/𝑇1 crossing at 𝜃𝑁𝑁𝑁 ≈ 1200. The 𝑇1 state remains more

energetically favourable than 𝑆0 over a small range of 𝜃𝑁𝑁𝑁 in this region,

Figure 3.5B. Thus, the bending mode of N3− appears to offer a mechanism for

the athermal electronic excitation of the molecule.

Discussion of the PES associated with δRS is done with respect to the

symmetric N-N bond lengths, Figure 3.5C. Across the eigenvector of this mode,

the T1 state remains lowest in energy amongst the excited states. In contrast

to the bending mode, however, extending the eigenvectors of this mode does

not lead to a CI, even up to a bond stretch of 2.0 Å and an associated ΔU ≈

11 eV. Similarly, discussion of the PES of δRA requires definition of a distortion

parameter α. This dimensionless value represents the degree to which the

eigenvector is perturbed, with R1 = Reqm + α/10 and R2 = Reqm − α/10 in

Figure 3.5D. Due to contraction of R2 as the eigenvector is imposed on

equilibrium geometries, the energy is found to rise considerably faster than for

the symmetric mode. Again, no CI is observed below Δ𝑈 ≈ 30 eV along this

eigenvector.

It follows from the above that a CI is only attainable through the bending motion

of N3− . However, the geometry of a real molecule results from the time-

dependent superposition of all vibrational normal modes. The combination of

δθNNN with δRS and δRA are therefore of interest. At relatively low energies,

107

the excitation energies between the So and excited states decreases

substantially further in δRS as compared to δRA, Figure 3.5. Hence, only the

combination of δθNNN + δRS is considered here. At θNNN = 1500 (Δ𝑈 ≈ 0.7

eV according to Figure 3.5B), the PES of δRS is found to deviate from that

observed in the linear molecule, Figure 3.6A in comparison to Figure 3.5C.

Most notably, with only a small distortion of θNNN , the T1/S0 CI becomes

accessible via δRS, with the CI observed at RS ≈1.65 Å. However, this pathway

is clearly not most energetically favourable. The total ΔU (i.e.

ΔU(δθNNN)+ΔU(δRS)) associated with this CI is over twice that required to

achieve the CI by bending alone. As θNNN is decrease to 130o (Δ𝑈 ≈2 eV

according to Figure 3.5B), it is instead possible to access the CI along the δRS

eigenvector (at RS ≈ 1.5 Å) with a total ΔU ≈ 4 eV, Figure 3.6B. The total

energy required to achieve the CI by this pathway is therefore comparable to

that required to access it by bending alone, but requires a smaller distortion of

the molecule within the confinements of a crystal lattice.

Figure 3.6: The PES for the symmetric stretch at (A) θNNN = 1500 and (B) θNNN = 1300. In each case

the potential energy surface for S0 (black), S1 (red), S2 (blue), T1 (pink) and T2 (green) are given.

The T1 state is accessible by extending the normal modes of the N3− molecule.

For the linear geometry in this state, the N3− dissociation barrier was found to

be ca. 1 eV. When θNNN is bent to 150𝑜, the S0 dissociation energy decreases

from 4.5 to 3.6 eV, with the dissociation barrier on the T1 PES decreasing to

0.67 eV, Figure 3.7A. As compared to the linear geometry, however, the

108

energetic drive to dissociation at this angle decreases considerably, and is only

-0.2 eV at this angle. At θNNN = 1500, the dissociation barrier on the S1 PES

also decreases, albeit minimally (from 1 eV to 0.88 eV). However, at this angle,

dissociation on the S1 surface is no longer energetically favourable. If θNNN is

reduced further to 1200 (i.e. the geometry of the CI), the barrier to dissociation

on the T1 PES remains approximately the same as at θNNN = 1500, 0.69 eV,

although that on the S0 surface drops drastically, from 3.6 eV to 1.6 eV, Figure

3.7b. The dissociation barrier on the T1 PES decreases to 0.34 eV at θNNN =

1100 , and is completely absent at θNNN = 1000 . At both 𝜃𝑁𝑁𝑁 =

1100 and 𝜃𝑁𝑁𝑁 = 1000 , dissociation on the T1 PES is overall exothermic,

Figure 3.7c-d. It follows that, near the T1/S0 CI, dissociation of N3− is more

accessible than under equilibrium, linear geometry.

The energies required to reach the CI for the pure azido anion are larger than

are generally considered attainable by measurement of temperatures under

mild impacts (typically < 3000 K). However, this energy translates into orders

of 1.5 eV/molecule, increasing with stronger impacts.15 Furthermore, it has

been shown that localisation of up-pumped energy in the region of defects can

be substantially higher,4 and sufficient to overcome bond dissociation

barriers.12 It is also worth mentioning that the periodicity of the crystalline state

leads to a reduction in the energy separation between ground and excited

states, particularly in the sensitive azide materials. Thus, smaller energies will

be required to achieve this excitation in these materials. The interaction of

cations with the azido anion in polymeric and molecular systems also reduces

the frequency of the bending vibrational mode. The ΔU associated with these

bends therefore decrease further. Critically, it is evident that electronic

excitation of the azido anion can be achieved by a purely mechanical route,

via the bending motion of the molecule.

109

Figure 3.7: Elongation of R2 for N3- with (A) θNNN = 1500, (B) θNNN = 1200, (C) θNNN = 1100

and (D) θNNN = 1000. In each case the potential energy surface for S0 (black), S1 (red), S2 (blue),

T1 (pink) and T2 (green) are given.

3.5.2 Metallisation in the Azides: Case Study of 𝛂-NaN3

To assess the validity of the gas phase calculations within the solid state, the

band structure was followed as a function of the normal mode eigenvectors

using an example energetic azide, α -NaN3. With discussion of external

vibrational modes, it is non-trivial to define the eigenvector as a function of an

internal coordinate. Instead, it is convenient to define a perturbative term

associated with ‘walking’ along each eigenvector,

𝑇𝑖 = 𝛼𝜖𝑖𝑅𝑒𝑞𝑚

Equation 3.2

110

where 𝛜𝐢 describes the normalised eigenvector of mode i that perturb the

equilibrium atomic position 𝑹𝒆𝒒𝒎 by a factor of α . Perturbations of the

eigenvectors were placed on the conventional cell, as it allows a more direct

calculation of the perturbation of neighbouring unit cells. The conventional cell

is the result of doubling the primitive cell, and hence halving the Brillouin zone.

As such, the conventional cell contains twice the number of vibrational

frequencies as the primitive cell, Table 3.5. The additional set of 12 modes

correspond to the edge of the primitive Brillouin zone and hence to

neighbouring primitive cells being directly out of phase. This is often the

maximum energy state of these vibrational bands.

Table 3.5: Calculated vibrational frequencies (ω) for the primitive and conventional unit cells of

α-NaN3 using PBE-D2.

Mode ω Primitive ω Conventional Assignment

M4 -- 84.78 Lattice

M5 -- 88.61 Lattice

M6 150.97 151.41 Lattice

M7 -- 154.47 Lattice

M8 -- 157.09 Lattice

M9 177.99 184.34 Lattice

M10 202.21 206.42 Lattice

M11 214.51 219.05 Lattice

M12 220.22 221.37 Lattice

M13 -- 221.54 Lattice

M14 -- 226.97 Lattice

M15 -- 230.22 Lattice

M16 -- 236.87 Lattice

M17 606.27 610.35 In phase δθNNN

M18 609.71 614.72 In phase δθNNN

M19 -- 617.49 Out of phase δθNNN

M20 -- 618.24 Out of phase δθNNN

M21 -- 1247.99 Out of phase δRS

M22 1250.43 1250.25 In phase δRS

M23 -- 1929.65 Out of phase δRA

M24 1959.81 1964.65 In phase δRA

111

Experimental data regarding the lowest frequency modes (i.e. lattice modes)

is sparse. To explore the validity of DFT to model lattice modes in ionic azides,

the inelastic neutron scattering spectrum (INS) of α-NaN3 was obtained, Figure

3.8. Unlike Raman and infrared spectroscopy, INS is not limited by quantum

selection rules, and therefore all vibrational modes are in principle visible. The

INS spectrum reveals the five lattice modes, the three highest with observed

frequencies ~220 cm-1, 210 cm-1 and 196 cm-1. The calculated frequencies

agree well with these experimental values, occurring at 220 cm-1, 214 cm-1 and

202 cm-1. Two additional features are observed at ~160 cm-1 and 130 cm-1 in

the INS spectrum, which appear to be somewhat lower in frequency than the

calculated frequencies of 177 cm-1 and 150 cm-1 using the D2 dispersion

correction. Low frequency vibrations are extremely sensitive to the weak

underlying potential of the surrounding crystal. The zone-centre vibrational

modes were therefore re-calculated using a second common dispersion

correction scheme, TS, which is somewhat less empirical than the D2 scheme.

The optimised primitive unit cell obtained under the DFT-TS scheme had a

volume ca. 1.2% below the experimental volume (as compared to DFT-D2,

which overestimated the volume by 1.1%). The frequencies that result from

the DFT-TS scheme show poorer agreement with the higher frequency lattice

modes (226 cm-1, 189 cm-1 and 187 cm-1), although it did lead to slight

improvements of the lowest frequency lattice modes (127 cm-1 and 174 cm-1).

The DFT-D2 scheme was therefore selected for further use. The internal

vibrational modes are modelled less accurately. The bending frequency (DFT-

D2) is calculated to be ~606/610 cm-1, ca. 4.5% lower than the measured INS

frequency of 639 cm-1. The symmetric stretching mode is modelled even more

poorly at 1250 cm-1, ca. 8% lower than the INS value of 1358 cm-1. However,

the calculated δ𝑅𝑆 frequency does agree well with previous simulations and

suggests an inherent inability of the PBE scheme to capture this mode.59 Note

that the band corresponding to δ𝑅𝑎𝑠 was not observed in the INS spectrum

likely due to the low scattering cross section of 14N and the low amplitude of

the asymmetric stretching mode. The calculated ν(δ𝑅𝑎𝑠) can therefore be

compared to literature Raman spectra.83 The frequency of δ𝑅𝑎𝑠 is better

112

reproduced by the PBE-D2 than δ𝑅𝑠, simulated to occur at 2037 cm-1 (1959.98

cm-1 without LO-TO correction) and the experimental Raman83 frequency at

2043 cm-1, a 0.2% underestimation). Overall, it therefore appears that the

PBE-D2 based scheme leads to a good correlation with experimental

frequencies in the external mode region and δθ𝑁𝑁𝑁. The latter is particularly

important as it is the target frequency identified in Section 3.5.1.1

Despite the agreement between zone-centre simulated low-frequency bands

and the INS spectrum, there are two striking differences:

• A well-defined band is observed at ca. 100 cm-1 in the INS spectrum

• The experimental intensities are poorly reproduced by simulation.

Both effects can be explained by noting that the TOSCA spectrometer does

not probe the Brillouin zone centre, Chapter 2.2.2.2, but spans a broad range

of momentum transfer.84 As scattering from both N and Na are dominated by

coherent scattering, vibration dispersion through the Brillouin zone becomes

important.

Despite the high frequency associated with the top external bands near 𝒌 = 0,

these frequencies represent only a small subset of the Brillouin zone, Figure

3.9. By comparison with the simulated INS spectra in Figure 3.8, it can be

inferred that these frequencies are diluted (e.g. by powder averaging85) as they

are not observed when simulated scattering from the full Brillouin zone is

considered. Instead, the highest feature observed in the simulated INS band

occurs at ca. 240 cm-1, consistent with the average frequency of these external

bands. This is only slightly higher than the experimentally observed highest

frequency (ca. 230 cm-1). This explains why the zone-centre calculation of α-

NaN3 without LO-TO correction offered a good starting point in Figure 3.8. In

fact, inclusion of the LO-TO correction for the zone centre calculation leads to

gross overestimation of the INS frequencies of the external modes.

113

Figure 3.8: Inelastic neutron spectra of α-NaN3 at 10 K obtained on the TOSCA spectrometer. (A)

The experimental spectrum (top) is shown alongside the simulated INS spectra (bottom) using

PBE-D2 (blue) and PBE-TS (green) methods. The INS spectrum is truncated at 1500 cm-1, as no

bands are observed above this frequency. No LO-TO correction is included in the simulated

spectra and only first order quantum events are included. (B) Modelling of the INS spectrum

based on the primitive cell, DFT-D2 phonon dispersion curve using different q-point sampling

densities.

114

Figure 3.9: Phonon dispersion curve calculated using PBE-D2 for the primitive cell of 𝛼-NaN3. The

zone centre frequencies calculated with LO-TO correction are highlighted as pink dots.

3.5.2.1 Band gap dependence on external lattice modes in 𝜶-NaN3

The DFT-D2 scheme leads to reliable calculation of the external vibrational

modes and hence can be used for further investigation. It is convenient to

begin with discussion of the external vibrational modes that contain no Na

character, M6, M8, M9 and M16 (Table 3.5). These four modes correspond to

tilting of the N3− molecules. In the first, azide molecules tilt in phase, polarized

primarily along the crystallographic b-axis. The second corresponds to a tilt

along this same axis, with each of the azido anions tilting out of phase with one

another. Hence, these modes correspond to the zone centre and primitive

Brillouin zone edge, respectively. Modes 𝑀9 and 𝑀16 describe the same

motion polarized primarily along the crystallographic a-axis.

As M6 and M8 are followed, the band gap is found to decrease dramatically (i.e.

towards metallisation), Figure 3.10A and 3.10B. However, this appears to be

an artefact of the rectilinear nature of the imposed eigenvectors. Indeed, if N-

N bond lengths are corrected to the equilibrium bond lengths, this trend

towards metallisation is lost. Only a small reduction in the bad gap is observed

at very large perturbations from the equilibrium geometry. The large difference

115

observed between rectilinear and corrected distortions clearly shows that the

addition of internal molecular modes may be promising to induce metallisation.

Figure 3.10: Tilting of the azido anion molecule showing (A) M6 (B) M8 , (C) M9 and (D) M16 .

Closed symbols are derived from rectilinear application of eigenvectors, and open symbols result

from correction to the N-N bond lengths. The band gap is initially a direct band gap (D), but

becomes indirect (I) on perturbation. Inset graphics show the distortion (green molecule) with

respect to the unperturbed (purple) molecular position. Band gap based on HSE06 calculation.

Along M9 , the band gap is seen to decrease very slightly before returning

towards its equilibrium band gap at higher distortions, Figure 3.10C. In stark

contrast, however, the band gap decreases markedly along M16, Figure 3.10D.

Imposing very large perturbations along this eigenvector (to a factor of 8)

decreases the band gap asymptotically to ca. 0.15 eV, but does not reach

metallisation, Figure 3.11. This is associated with ΔU ≈ 5.6 eV.

116

Figure 3.11: Calculated band gap for 𝑀16 extended to large perturbation. Band gap based on

HSE06 calculation.

On further analysis of the band structure, it is found that this band gap

narrowing does not result from a closing of the N3− band gap, but rather from a

closing of the N…Na band gap, Figure 3.12. A comparison of the absolute

energies of the valence bands suggests that this results from an increase in

the N valence band energies by approximately 3.5 eV, which occurs as the

occupied non-bonding πg orbitals of the azido anions are forced together.

While band gap narrowing in this manner would not permit excitation of the N3−

molecule, it may permit transient oxidation of N3− → N∙

3 + e−. The process77

2𝑁3− + 2𝑁𝑎+ → 2𝑁3

∙ + 𝑁𝑎

2𝑁3∙ → 3𝑁2

has been suggested as a possible thermal decomposition mechanism of the

azides. Band gap narrowing by 𝑀16, however, results in an indirect band gap.

The rate of excitation across such band gap transitions are very slow.86

Moreover, given that the narrow band gap exists across only a small subset of

k-space, Figure 3.12, very few potential excitation channels are available. With

the impact-induced vibrational energy transfer processes occurring on the sub-

nanosecond (picosecond for phonon-phonon dissipation), it is reasonable to

117

suggest that electronic excitation along eigenvector M16 are simply too slow to

be considered here. However, this does suggest a potential mechanism for the

thermally induced decomposition of these materials, with long-duration

excitation of the lattice. While the exact rationale governing the inactivity of

M16 requires further investigation, experiment has demonstrated that α-NaN3

is not reactive to impact initiation.30 It therefore follows that M16 in unlikely to

be responsible for impact-induced initiation.

Figure 3.12: Comparison of the electronic band gap under equilibrium and after imposing an α =

8 fold perturbation of M16. (A) The partial DOS is given for (black) N and (blue) Na species. The

vertical dotted line on (A) is given to indicate the Fermi surface (ϵF), and the red vertical line

shows the energy of ϵF relative to the unperturbed structure. (B) The electronic band structure

plotted along high symmetry lines in the Brillouin zone for (left) unperturbed and (right) α = 8

structures.

118

The remaining nine external modes all exhibit a mixture of N and Na

displacement. It is reassuring to find that none of these modes lead to any

notable decrease in the band gap, Figure 3.13. There is typically no more than

a ca. 1 eV decrease in the band gap along any of these eigenvectors, with M12

(the out of phase translation of Na and N3− species along the crystallographic

c-axis), leading to an overall increase in the band gap. The only exception is

M15, which leads to rapid decrease in band gap energies at large α. However,

it must be noted that this is associated with a 60 eV increase in energy, which

arises due to the eigenvector contracting the distance between neighbouring

Na+ ions.

Figure 3.13: Effect of external vibration normal coordinates on the energies and band gap of

αNaN3. Mode numbers are indicated in brackets in each plot. Modes are identified as, (Left) M4

(-●-), M5 (-■-), M7 (-▲-), M10 (-♦-), (Right) M11 (-■-), M12 (-●-), M13 (-▲-), M14 (-♦-), M15 (-

★-).

3.5.2.2 Band gap dependence on internal vibrational modes in 𝜶-NaN3

The subsequent four vibrational modes (M17 - M20) correspond to the δθNNN

modes. M17 and M18 are the zone-centre (i.e. in phase) modes, perpendicular

to and along the crystallographic b-axis, respectively. M19 and M20 are their

corresponding out-of-phase modes.

119

Figure 3.14: Effect of M17 (left) and M19 (right) on the band gaps and energies of α-NaN3. Band

gap momentum conservation is indicated as direct (D) or indirect (I). The arrow indicates

continuation of an indirect band gap. Band gaps from HSE06 calculation. To reflect perturbation

of two azido anions (in the conventional cell), energy is given per molecule.

To follow the band gaps associated with 𝑀17 - 𝑀20, the rectilinear perturbation

was applied according to Equation 3.2, and the N-N bond lengths were

restored to equilibrium lengths. This was done to ensure only the isolated

normal coordinates were investigated. As the azido anion was perturbed along

the bending mode, 𝛿𝜃𝑁𝑁𝑁, the band gap was found to decrease steadily with

angle. The band gap reaches approximately half of its original value at θNNN ≈

130o, and continues to decrease on further bending. By 110o, the band gap

drops to 0 eV, and the material is found to have metallised, Figure 3.14. This

metallisation is of particular interest as it corresponds to a crossing of the 𝑆0/𝑆1

PES, which was not observed in the isolated gas-phase molecule. This can be

suggested to result from the band gap narrowing that occurs when a molecule

is introduced into a periodic crystal.74 Note as the calculations performed here

were single-reference ground state, closed shell simulations, the triplet states

observed in the multi-references CI calculations (Section 3.5.1.1) were not

considered here. However, given that the multi-reference calculations suggest

that the T1 state should exist ca. 1 eV lower in energy than the S1 state, it is

reasonable to propose that the T1/𝑆0 CI may be accessible in the crystalline

120

lattice model at θNNN ≈ 115 − 1200, and Δ𝑈 ≈ 4-4.5 eV.molecule-1. As both

the in- and out-of-phase modes exhibit the same behaviour modes 𝑀18 and

𝑀20 are not reported here.

Unlike for M16, the metallisation that is observed along M17 − M20 is not the

result of a decreasing N…Na band gap. Instead, it occurs by a decrease in the

N3− conduction/valence band gap. While comparison of the absolute energies

does suggest partial increase in the energy of the valence band as a function

of this bend, the major effect results from a lowering in the conduction band

energies, Figure 3.15. While the band gap is again indirect, it is worth noting

how flat the band gap is compared to Figure 3.12 and therefore the existence

of many more available transition channels. Thus, a potential 𝑆0/𝑆1 CI exists

in the solid state and again permits excitation of the azido anion. The lowering

of the conduction band to such considerable degrees also offers a role for local

electronic defects within these structures (e.g. holes or dopant states), which

sit within the band gap of the pure crystalline material. These defects have

previously been suggested as being crucial for the initiation of energetic

compounds, although no mechanism for their athermal influence has been

proposed.71,87 It can be suggested that their interaction with the electronic

structure, and its dynamics as a result of normal mode perturbation, may be

crucial to understanding their mechanism of action.

121

Figure 3.15: (A) Partial electronic density of states as a function of 𝑀17, with contributions from

N3- (black) and Na+ (blue). The relative energy of ϵF at θNNN=180o is shown as a red vertical line

in the perturbed structures. (B) Electronic band structures for NaN3 with θ𝑁𝑁𝑁 = 180𝑜 and

θ𝑁𝑁𝑁 = 110𝑜. ϵ𝐹 is marked with a dotted line.

122

Figure 3.16: Effect of 𝑀22 and 𝑀24 (i.e. δ𝑅𝑠 and δ𝑅𝑎𝑠, respectively) on the electronic band gap.

The latter is plotted as α, defined as in Figure 3.5. Band gaps are defined as direct (D) or indirect

(I) and the arrow indicates that the indirect band gap continues. To reflect perturbation of two

azido anions (in the conventional cell), energy is given per molecule

M21 is the out of phase δRS mode. Due to the contraction of one set of N-N

bonds over this normal coordinate, it was not possible to follow this mode

beyond a N-N stretch of 1.4 Å. By this limit, the band gap was found to

decrease only slightly, from ca. 5.2 to 3.3 eV. It is therefore unlikely that this

mode provides a route to metallisation. The in-phase (zone centre) δ𝑅𝑆 is

expressed as M22. By the same Δ𝑈 (ca. 5 eV.molecule-1) at which the bending

modes led to metallisation, the band gap from δRS reduces to only ca. 2 eV.

Further extension of this normal coordinate unsurprisingly leads to

metallisation as the N-N bonds are ruptured. However, this occurs at ΔU > 12

eV.molecule-1. The final two normal coordinates, M23 and M24, are the in- and

out-of-phase δRas , respectively, and behave the same. For the reason

described for M21, there is a physical limitation on the maximum Ti that can be

applied to these modes. This limit was reached with an energy penalty of ca.

13 eV.molecule-1, by which point the band gap was found to reduce to only 3.8

eV. Thus, metallisation can only be attained via the bending normal coordinate,

consistent with findings for the gas phase N3− molecule. The solid state,

however, introduces the availability of a S1/S0 CI, which is not apparently

available in the gas phase.

123

Thus to summarise, two sets of modes have been identified that lead to a

narrowing of the band gap in α-NaN3. The first is a phonon mode, M16 which

suggests that a route to N3− → N3 might be possible under the application of a

long-duration vibrational excitation but does not occur under mechanical

impact. Modes M17 − M20 support the gas-phase calculations, identifying the

bending mode as being a probable target mode for initiation of explosions in

azide materials.

3.5.3 Up-Pumping and Impact Sensitivity

Note the phonon dispersion curves for Sn(N3)2, NH4N3, TAGZ and HN3 were calculated by Dr. Carole

Morrison (School of Chemistry, University of Edinburgh)

The discussions presented in Sections 3.5.1 and 3.5.2 suggest that vibronic

processes may be responsible for the spontaneous electronic excitation of the

explosophoric N3− species. These processes are driven by the normal

coordinate eigenvector of δθNNN, but require perturbations that are larger than

are typical under thermal equilibrium. Hence, to reach the CIs that appear

along the PES of N3− , the molecule must be promoted to a highly excited

vibrational state. This can be achieved by phonon up-pumping. This process

is given in Equation 3.3, which describes the vibrational lifetime of a mode with

branch index j and wave vector, 𝒒,88

γ𝐪,j =𝜋

ℏ2𝑁𝑞∑ |𝑉

𝒒𝑗,𝒒′𝑗′,𝒒"𝑗"

(3)|2

𝒒′,𝑗′,𝑗"

× [(1 + 𝑛𝒒′𝑗′ + 𝑛𝒒"𝑗")𝛿(𝜔𝒒𝑗 − 𝜔𝒒′𝑗′ − 𝜔𝒒"𝑗")

+ 2(𝑛𝒒′𝑗′ − 𝑛𝒒"𝑗")𝛿(𝜔𝒒𝑗 + 𝜔𝒒′𝑗′ − 𝜔𝒒"𝑗")]

Equation 3.3

Equation 3.3 restricts discussion to within the first anharmonic approximation.

This is a reasonable restriction as higher order terms occur too slowly in most

cases.4 The phonon lifetime can be understood by two sets of scattering

processes, which are displayed in the square brackets. The first term

describes the down-conversion process, where vibration 𝜔𝒒𝑗 decomposes into

two lower-frequency modes, 𝜔𝒒′𝑗′ and 𝜔𝒒′′𝑗′′. The second term describes the

124

combination of two phonons, 𝜔𝒒𝑗 and 𝜔𝒒′𝑗′, to create a third, 𝜔𝒒′′𝑗′′. Where

𝜔𝒒′′𝑗′′ > 𝜔𝒒𝑗, this process is known as up-conversion. At finite temperature,

the scattering processes described in each event depend on the Bose-Einstein

statistical occupations (nq,j), Equation 3.4, and a third-order anharmonic

coupling constant,𝑉(3) . The magnitude of the latter term depends on the

relative polarisation and anharmonic character of the three coupling phonon

modes. As both up- and down-conversion processes are possible, excess

energy is rapidly equilibrated through the molecule via the available vibrational

relaxation mechanisms. Thus, to achieve a highly excited state of a target

mode (δθNNN in the case of the azides) it is important to achieve rapid

conversion into the corresponding branch, j. The slower the conversion into

the branch, the more the required input energy to achieve sufficient excitation.

nω = [𝑒(ℏ𝜔/𝑘B𝑇) − 1]−1

Equation 3.4

It follows from Equations 3.3 and 3.4 that energy transfer rates will be faster

when including low frequency modes, which at temperature, 𝑇, will exhibit

higher populations, and are typically more anharmonic.4 As described in

Section 3.1, a mechanical impact can be treated as instantaneous heating of

the lowest frequency vibrational modes.12 This therefore leads to highly

populated phonon states, which rapidly reach a quasi-equilibrium state. For

simplicity, the model employed here chooses this initial equilibrated phonon

bath as a starting point.

At this starting point, it is convenient to construct a temperature-independent

model, by extending Equation 3.3 to the low temperature limit of T = 0 K.

Under this limit,

γ𝐪,j =𝜋

ℏ2𝑁𝑞∑ |𝑉

𝒒𝑗,𝒒′𝑗′,𝒒"𝑗"

(3)|2× [𝛿(𝜔𝒒𝑗 − 𝜔𝒒′𝑗′ − 𝜔𝒒"𝑗")]

𝒒′,𝑗′,𝑗"

Equation 3.5

125

Here the bracketed term represents the two-phonon density of states, Ω(2). In

the absence of temperature considerations, microscopic reversibility dictates

that the number of down-conversion pathways must equal the number of up-

conversion pathways. Hence, Equation 3.5 describes the total number of

scattering pathways that can transfer energy into mode 𝜔𝒒𝑗. In this form, 𝜔𝒒𝑗

is defined as the target frequency (herein labelled 𝜔𝑻, the N3− δθNNN mode),

with 𝜔𝒒′𝑗′ and 𝜔𝒒"𝑗" denoting lower frequency modes. The Dirac 𝛿 ensures

conservation of energy, and momentum is conserved by setting 𝒒 = −𝒒′ − 𝒒′′.

Energy transfer to 𝜔𝑻 is then largely dependent on the number of pathways

defined by Ω(2).

By imposing the Einstein approximation (that ω is 𝐪 -independent) for the

internal vibrational modes, it is possible to consider only the phonon density of

states (PDOS), rather than the full phonon dispersion curves. The latter are

shown in Figure 3.17. This is based on the following:

1. To a good approximation, there is a continuum of vibrational states

within the phonon bath.

2. For phonon-phonon coupling involving two phonons with the same

branch index, j1 = j2 , momentum conservation and phase matching

dictate that 𝒒𝟏 = −𝒒𝟐, such that ωj,𝐪 + ωj,−𝐪 = 𝜔𝑇 ,𝒒=0. Thus, 𝜔𝑇 is only

accessible at the zone centre and only the absolute frequency of the

low frequency modes is important.

3. For phonon-phonon coupling involving two phonons with different

branch index, j1 ≠ j2, the 𝐪-independence of ωT imposes that for any

combination of (𝜔𝒒𝑗, ωT) , there will be a 𝜔𝒒′𝑗′ at the appropriate

momentum and frequency to satisfy Equation 3.5. The same holds

under the assumption that one of the phonon modes is a doorway mode,

regardless of its 𝐪-dependence.

4. In the absence of explicit consideration of 𝑉(3), coupling between all

sets of phonons can be taken to be approximately the same, provided

they comprise distortion of the same set of interacting atoms (i.e. are of

the same molecule or strongly bonding intermolecular atoms).89,90

126

Figure 3.17: Phonon dispersion curves for the energetic azides. All are given for the primitive cell except AgN3.

127

3.5.3.1 Partitioning of the Vibrational Structure

Following from the three terms contained in Ω(2) of Equation 3.5, the PDOS

can be segmented into a series of physically meaningful regions, Figure 3.18.16

The first mode, 𝜔𝒒′𝑗′, generally exhibits lattice character, and is held within the

phonon bath, which has an upper limit defined by Ωmax. While this value is not

rigorously defined, it can be qualitatively described as the highest lattice-based

mode. Due to the high anharmonicity of these lattice modes, and the high

density of vibrational states, the thermalisation of vibrational energy occurs

very rapidly in this region. This imposes a crude definition of Ω𝑚𝑎𝑥 as being the

first point in which the phonon density of states drops to zero. The second

frequency, 𝜔𝒒"𝑗", generally sits somewhere between Ω𝑚𝑎𝑥 and 2Ω𝑚𝑎𝑥, and is

termed the ‘doorway mode’. The upper limit of 2Ω𝑚𝑎𝑥 is significant as it defines

the highest frequency attainable by coupling of two phonon bath modes. The

third mode, 𝜔𝑇 is the target vibrational frequency. It must fall within 3Ω𝑚𝑎𝑥 to

be accessed by coupling of a phonon mode with a doorway mode. The identity

of 𝜔𝑇 was determined in Section 3.5.1 as being 𝛿𝜃𝑁𝑁𝑁. Within the Einstein

approximation, the frequency of 𝛿𝜃𝑁𝑁𝑁 can be identified from the Γ-point

eigenvectors. This was done by artificially extending the eigenvectors of each

zone-centre normal coordinate and led to easy identification of 𝜔𝑇, Figure 3.19.

Note that due to factor group splitting and symmetry independent azido anions,

multiple distinct δθ𝑁𝑁𝑁 frequencies can exist in the same crystal. The covalent

compounds exhibit a more extensive spread in 𝛿𝜃𝑁𝑁𝑁 frequencies. While a

cluster of δθNNN exists around 600 cm-1 in all of the azides (with the exception

of HN3), the covalent (more sensitive) compounds exhibit additional δθNNN at

lower frequency. Within the framework of the up-pumping model of impact

initiation, this may be indicative of their higher sensitivity and offer a design

target for novel materials.

128

Figure 3.18: The phonon density of states for the crystalline azide materials studied here. The

vertical dotted line indicates Ωmax, and the blue rectangle highlights the position of ωT. Red

indicates azido anion partial DOS Figure is adapted from Ref 21.

129

Figure 3.19: Identification of target modes. The azide bend at each Γ-point normal mode is

highlighted.

The position of Ωmax is important as the anharmonicity of ω < Ωmax leads to a

rapid equilibration of the population of these modes in accordance with both

up- and down-conversion processes of Equation 3.3.4 Experimental work has

130

suggested that this equilibration occurs at least an order of magnitude faster

than any up-conversion beyond Ωmax.4 The up-conversion of this energy then

occurs in two stages:

1. coupling of two modes with ω𝑗 < Ω𝑚𝑎𝑥 to excite a mode with Ω𝑚𝑎𝑥 <

ω < 2Ω𝑚𝑎𝑥, and

2. further up-pumping to modes with 2Ω𝑚𝑎𝑥 < ω < 3Ω𝑚𝑎𝑥.17

It is from this sequence that vibrational modes with Ω𝑚𝑎𝑥 < ω < 2Ω𝑚𝑎𝑥 derive

their name: doorway modes. While step (1) is first, step (2) occurs only

picoseconds afterwards17 and hence these processes may become important.

This sequence of steps also limits primary up-pumping steps to a maximum of

3Ω𝑚𝑎𝑥.

The probability (℘) of phonon-phonon coupling processes is governed by

Fermi’s Golden rule,91

℘(𝑖 → 𝑓) ∝ |⟨𝜑𝑓|𝐻3|𝜑𝑖⟩|2𝐷𝑓(𝐸)

Equation 3.6

where Df(E) is the density of final states and H3 is the third order anharmonic

Hamiltonian. Thus, the probability of scattering is a maximum when the initial

and final scattering states, |𝜑𝑖⟩ and |𝜑𝑓⟩ , respectively, are coherent.14

Qualitatively, it follows that the greater the total change in the PDOS, the less

probable the transition will be. Within the nomenclature introduced above, it

might therefore be expected that energy transfer to 𝜔𝑇 will occur more quickly

given a smaller ∆𝜔 = 𝜔𝑇 − Ω𝑚𝑎𝑥 . This is because more combinations of

phonon modes will be resonant with ω𝑇 as Δω → 0. Generally, more sensitive

materials are found to exhibit higher Ωmax values, and analysis of Δω does

suggest some merit to this qualitative approach, Table 3.6, although

discrepancies do arise. This is most notable with LiN3 and Ba(N3)2, which

appear in different sensitivity classifications according to this method.

131

Table 3.6: Characteristic frequencies for the azide materials. The Γ-point based target frequency

(𝜔𝑇), top of the phonon bath (Ω𝑚𝑎𝑥) and Δ𝜔 = 𝜔𝑇 − Ω𝑚𝑎𝑥.

Material 𝝎𝑻(Г) /cm-1 𝛀𝒎𝒂𝒙 /cm-1 ∆𝝎 /cm-1

NaN3 615 250 365

TAGZ 595 260 335

NH4N3 605 310 295

LiN3 605, 620 460 145, 160

HN3 435-500 225 210-275

Ba(N3)2 600, 615 265 335, 350

AgN3 581, 594 320 261, 274

Zn(N3)2 547, 630, 670, 685 445 102, 185, 225, 240

Sn(N3)2 550, 615 395 155, 220

3.5.3.2 Coupling Pathways and Impact Sensitivity

Within the first anharmonic approximation, only two phonons (𝜔𝑗′ and 𝜔𝑗′′)

may scatter to form a third ( 𝜔𝑇 ). This leads to two different scattering

mechanisms:

1. 𝜔𝑗′ and 𝜔𝑗′′ share the same branch index and frequency, i.e. 𝜔𝑗′ = 𝜔𝑗′′,

imposing the restriction that q(𝜔𝑗′) = - 𝒒(𝜔𝑗′′), and q(𝜔𝑇) = Γ.

2. 𝜔𝑗′ ≠ 𝜔𝑗′′ such that q(𝜔𝑇) = 𝒒(𝜔𝑗′)+ 𝒒(𝜔𝑗′′)

These two scattering mechanisms are analogous to spectroscopic processes

of (1) overtone and (2) combination bands. This terminology will therefore be

adopted for ease of the following discussion.

While explicit solution of Equation 3.5 requires the calculation of 𝑉(3) , its

calculation is intractable for large, low symmetry systems.88 Previous attempts

at deriving values of this term have suggested that structurally similar materials

exhibit minimal difference in the average value of 𝑉(3), and that its neglect is

generally sufficient.19,20,89 However, it follows from the definition of this term

that vibrational eigenvectors that do not comprise the same atoms will not

couple with any notable efficiency. This is an important consideration for

NH4N3 and TAGZ, where large portions of the vibrational structure contain no

132

N3− character. For these systems, coupling pathways are therefore considered

only for the azide-channel PDOS, Figure 3.18.

Overtone Pathways

In accordance with Equation 3.5, overtone pathways can be expected to occur

more efficiently.17 This assumption formed the base for previous work at

understanding impact sensitivity.14,19 However, the number of overtone

pathways is far fewer than the pathways available by the combination

mechanism. Further, only the first overtone pathway can be considered within

the first anharmonic approximation. Higher order overtones become

increasingly improbable, making these pathways less likely for materials in

which ω𝑇 > 2Ω𝑚𝑎𝑥. This restriction affects NaN3, TAGZ, and BaN3. However,

due to the high anharmonicity of phonon modes, it has been suggested that

quartic terms can occur with sufficient speed for further consideration. 93 This

extends the restriction to 3Ωmax.

To describe the overtone pathways, the PDOS, g(ω) is scaled by N, the

overtone number,

𝑔𝑁(𝜔) =𝑔(𝜔)

𝑁 ; 𝜔𝑁 = 𝜔𝑁

Equation 3.7

Within this model, the number of pathways through which energy can transfer

can then be taken as an integration of 𝑔𝑁(𝜔) at each 𝜔𝑇 , Figure 3.19. To

account for the existence of resonant vibrational states, a sampling window of

𝜔𝑇 ± 10 cm-1 was used. This reflects the magnitude of Gaussian smearing that

was applied during generation of the PDOS.

The number of overtone pathways present in each material does appear to

correlate loosely with the relative sensitivities, Figure 3.19B. As expected,

modes in which ωT > 2Ωmax have 𝑔2(𝜔𝑇)=0. The total 𝑔𝑁(𝜔𝑇) observed for

the insensitive materials is consistently less than for the sensitive materials,

with a sensitivity ordering following TAGZ < HN3 < NaN3 ≈ NH4N3 < Ba(N3)2 <

AgN3 < Sn(N3)2 < Zn(N3)2 for the first two overtones (i.e. quartic coupling).

133

While many of the materials are correctly ordered, the overtone pathways

alone are clearly incapable of fully describing the sensitivity ordering of these

materials. This deficiency is particularly notable for the low sensitivity materials

and HN3. However, in developing the up-pumping model, interest rests in the

localisation of energy. Hence, it is worth recasting these values based on the

number of molecules present in the unit cell (this also has the effect of

correcting for the use of conventional cells in some cases), Figure 3.19C.

Renormalisation in this manner highlights even further the deficiencies of

considering only overtone pathways in the up-pumping model, with HN3 in

particular being substantially underestimated in its sensitivity. The general

trend remains with sensitive compounds exhibiting higher integrated overtone

densities than the less sensitive compounds.

134

Figure 3.19: Overtone pathways for phonon up-pumping. (A) Example of overtone coupling in

AgN3, showing the (black) PDOS, (blue) N=2 overtone, (green) N=3 overtone and (pink) N=4

overtone. (B) Integration over 𝜔𝑇 ± 10 cm-1 for the overtone pathways available in the crystalline

azide materials, arranged in approximate order of increasing sensitivity. Values are given as sums

across all target modes. Overtones N=2 (black), N=3 (red) and N=4 (blue) are shown, alongside

N2+N3 (green), as well as N2+N3+N4 (orange). (C) Renormalisation of the N2+N3 overtone curve

by the number of N3− molecules in the unit cell.

135

Multiphonon Density of States: Combination pathways

Given the deficiency of the overtone pathways alone, it was necessary to

consider also the combination pathways. This was done by generation of Ω(2)

for each of the materials under investigation, Figure 3.20. By Equation 3.5,

only up-conversion processes are accounted for, ensuring 𝜔𝒒′𝑗′ and 𝜔𝒒"𝑗" <

𝜔𝑻 . To further adhere to the model proposed in Section 3.5.3, further

constraints are imposed on generation of Ω(2), ensuring that 𝜔𝒒′𝑗′ and 𝜔𝒒"𝑗" <

2Ω(2) and 𝜔𝒒′𝑗′ < Ω𝑚𝑎𝑥 . This ensures that all values of Ω(2) < 3Ωmax must

include at least one phonon mode. It is generally found, however, that this

restriction has little effect on the structure of Ω(2)( 𝜔𝑻), Figure 3.20. Generally,

it is found that 𝜔𝑻 falls within the region of this restricted Ω(2)for the sensitive

materials, with little to no Ω(2) density found at 𝜔𝑻 for insensitive materials. The

magnitude of Ω(2) is seen to increase notably with increasing sensitivity. Only

one exception (Ba(N3)2) is found to this trend. Despite its similarity to the PDOS

of AgN3, the lower value of Ωmax for Ba(N3)2 means that the 𝜔𝑇 sits just beyond

the doorway region. Thus, the magnitude of Ω(2) in the restricted case is

necessarily zero, given no doorway modes are present. The calculated Г-point

Ω𝑚𝑎𝑥 of Ba(N3)2 agrees well with experimental measurements (230-240cm-1)92

and suggests minimal error in the selection of the phonon bath for this material.

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Figure 3.20: Two phonon density of states (2PDOS; Ω(2)) for the azides, calculated by enforcing

both 𝜔𝒒′𝑗′ and 𝜔𝒒"𝑗" < 2Ωmax (black). The difference between this restricted 2PDOS and that

produced by considering all combination modes is also shown (blue). The PDOS is given in red.

Ω𝑚𝑎𝑥 is given as a vertical dotted line, and 𝜔𝑻 is indicated with an arrow.

137

Prediction of Impact Sensitivity

By imposing the approximation that ω𝑇 is flat (i.e. 𝒒-invariant), the total value

of Ω(2)(ωT) is indicative of the number of coupling pathways capable of up-

converting energy into this mode. Based on Equation 3.5, it follows that the

faster energy can transfer into ωT, the lower the dissipation of this energy and

the more sensitive will be the compound.

A clear correlation is observed between Ω(2)(ω𝑇) and the experimental impact

sensitivity for each compound, Figure 3.21. The insensitive materials exhibit

Ω(2)(ω𝑇) ≈ 0. This suggests that within the ideal crystal, direct transfer of

energy into ωT is not possible, and therefore that they cannot be easily initiated

by impact. The calculated value of Ω(2)(ω𝑇) increases with increasing

experimental impact sensitivity. However, by enforcing 𝜔𝒒′𝑗′ and 𝜔𝒒"𝑗" <

2Ω𝑚𝑎𝑥 and 𝜔𝒒′𝑗′ < Ω𝑚𝑎𝑥, Ba(N3)2 appears to be as insensitive as α-NaN3,

Figure 3.21A. If this restriction is lifted and consideration is therefore given to

all combination modes (𝜔𝒒′𝑗′ and 𝜔𝒒"𝑗" < 𝜔𝑻 ), the predicted sensitivity of

Ba(N3)2 increases in line with the sensitive materials, Figure 3.21B. It is worth

mentioning that the slope of Ω(2) for Ba(N3)2 is very steep; small changes in

the target frequency or integration window can therefore have considerable

influence on the resulting sensitivity prediction. The same is not true for the

other azide materials investigated here. In the frame of experimental

consideration, this suggests that the presence of defects, or the compression

that is associated with a shock wave, may drastically alter the sensitivity above

the 0 K levels predicted here. Similar phenomena are known.43

Unfortunately, despite yielding excellent correlation with experiment, it is not

sensible to lift the restriction of 𝜔𝒒′𝑗′ , 𝜔𝒒"𝑗" < 2Ω𝑚𝑎𝑥 and 𝜔𝒒′𝑗′ < Ω𝑚𝑎𝑥 based

on the model outlined in Section 3.5.3. This poses a problem for the relative

sensitivity of Ba(N3)2, which must abide by the same physics as the remaining

materials. As a final step it is therefore worth reintroducing the possible energy

coupling pathways available through overtones, with the second overtone

(N=3) being taken as the highest contributing overtone pathway, as discussed

138

above. If these pathways are considered, the predicted ordering becomes very

promising, Figure 3.21C-D. In the final ordering, Ba(N3)2 is predicted to be

slightly less sensitive than AgN3. LiN3 is found to sit at the interface between

the low and high sensitivity compounds. Both of these predicted orderings are

in agreement with experiment. Based on the results in Figure 3.21C, the

predicted sensitivity ordering of the azides follows as NaN3 ≈ TAGZ < NH4N3

< LiN3 < Ba(N3)2 < AgN3 < HN3 < Sn(N3)2 < Zn(N3)2. This appears to be

consistent with experimental reports.

Figure 3.21: Integrated Ω(2)(ωT) for the azides. (A) Based on Ω(2) generated under the restriction

of 𝜔𝒒′𝑗′ and 𝜔𝒒"𝑗" < 2Ωmax and 𝜔𝒒′𝑗′ < Ω𝑚𝑎𝑥 . (B) Based on Ω(2) with 𝜔𝒒′𝑗′ and 𝜔𝒒"𝑗" < 𝜔𝑻. (C)

Recasting of (A) with addition of 𝑔𝑁(𝜔𝑇) for N=2,3; and (D) recasting of (B) with addition of

𝑔𝑁(𝜔𝑇) for N=2,3.

139

As with overtone modelling, it becomes necessary to consider localisation of

this energy per molecule, Figure 3.22. When this is done, the same general

trend is observed as in Figure 3.21C, although the scale is recast. Only a

change in the ordering of HN3 and AgN3 is observed. Based on the molecule-

normalised up-pumping rates in Figure 3.22B, the final ordering is therefore

predicted as NaN3 ≈ TAGZ < NH4N3 < LiN3 < Ba(N3)2 < HN3 ≈ AgN3 < Sn(N3)2

< Zn(N3)2.

Figure 3.22: Recasting of Figure 3.21 (C-D) normalising by the number of azido anions in the unit

cell.

As highlighted in Section 3.3, the ordering of impact sensitivities of the

energetic azides is widely debated in literature. For example, the relative

ordering of Ba(N3)2 and AgN3 is contested, with most recent reports suggesting

AgN3 > Ba(N3)2.36 The sensitivity of Zn(N3)2 is also inconsistently reported in

literature, with some reports stating that is more sensitive than Pb(N3)2, acting

as a sensitizer.45 It is likely that these discrepancies result from variations in

experimental conditions (particle size, crystallinity, impurities, etc). However,

the model presented here does appear to correlate well with the majority of

literature reports. This model may therefore be helpful in clarifying the ordering

140

of azide materials, or highlight areas for deeper experimental investigation,

where reports are drastically different.

3.6 Conclusions

Azides represent a broad class of energetic compounds, covering a wide

range of structural chemistry and impact sensitivity. To understand the

initiation of a chemical explosion, it is necessary to understand the

decomposition of the explosophoric moiety. For the simple azides, this is

rupture of an N-N covalent bond. The electronic structure of the N3− molecule

was therefore investigated. Bond dissociation was found to be possible via an

athermal mechanism, provided sufficient excitation of the bending vibrational

mode of N3− . At sufficient perturbation of the nuclear geometry, a conical

intersection (CI) was observed between the S0 and T1 electronic states. The

potential energy surface of the latter readily facilitates N-N bond dissociation.

The existence of this CI was verified in the solid state.

Noting that mechanical impact leads to rapid excitation of lattice vibrations, the

up-pumping model was explored as a mechanism for impact initiation. Based

on the electronic structure calculations, the bending motion was selected as

the target vibration into which vibrational energy must up-convert to initiate an

explosion. Considering both overtone and combination up-conversion

pathways, it was found that the relative rate of up-pumping into this target

vibration led to excellent correlation with experimental impact sensitivities.

Thus, the work presented in this chapter demonstrates the first fully ab initio

approach to the prediction of the relative impact sensitivities of energetic azide

materials, without the need for any empirical correlations.

The rate of up-conversion was found to be largely dependent on two key

vibrational frequencies: 1) the maximum phonon frequency, Ω𝑚𝑎𝑥, and 2) the

frequency of the N3− bending mode, 𝜔𝑻 . This therefore offers targets and

rationales for the design of novel materials:

141

1. Ω𝑚𝑎𝑥. This value depends on the nature of the external lattice modes

and crystal packing. As such, a model based on vibrational energy

transfer includes potential for understanding the different sensitivities of

polymorphic and multi-component materials (co-crystals and salts).

Stronger bonding of the N3− anion within the lattice (i.e. polymeric or

molecular structures) was found to correlate with higher Ω𝑚𝑎𝑥. It also

follows that more compressible materials will exhibit a higher Ω𝑚𝑎𝑥

when subject to a mechanical perturbation. As such, materials based

on weaker non-covalent interactions between energetic molecules may

be more sensitive

2. 𝜔𝑇. This value depends on the bonding nature of the N3− anion within

the crystal lattice. Higher covalent character leads to a decrease in 𝜔𝑇,

and thus enhanced sensitivity. The increased covalent character

between the N3− molecule and a cation also appear to weaken the N-N

bond. Initiation may therefore be easier.

A particular strength of the model presented in this chapter is the fact that it

encompasses many aspects of earlier models reported in the literature. For

example, within the framework of this chapter, there are clear rationales for a

correlation of band gap and bond dissociation energy with impact sensitivity.

Further, effects such as packing density and crystal packing can all find a

physical basis within this model, via compressibility and the capability of

molecules to undergo necessary geometric perturbations. This work therefore

makes strides towards an overarching understanding of initiation in energetic

materials.

Despite its promise, the present model is based on idealised crystalline

structures, which do not exist in reality. The model therefore only offers insight

into the intrinsic potential of a material to initiate under mechanical perturbation.

Further work will be required to introduce non-ideal features, such as defects

and surfaces. Further, while the model here can justify the relative sensitivity

of energetic materials, it is not yet able to determine whether a material will be

energetic in the first place. This is to say, the model describes the propensity

142

of a material to react under mechanical perturbation, but it does not determine

what that reaction will be. That aside, the present contribution offers a powerful

platform from which novel materials can be designed in silico and offers novel

insight into the structure-property relationships of common energetic materials.

3.7 Suggestions for Future Work

The material presented in this chapter clearly identifies the up-pumping model

as a powerful tool to understanding the impact sensitivity of the crystalline

energetic azides. This offers a starting point for numerous follow-up

investigations.

• The decomposition pathway of the azido anion within the crystal structure

would offer important validation of this model. While the CI was identified

in the 𝛼 -NaN3 lattice, it would be interesting to investigate how this

translates across the azides. In particular, how the PES associated with the

bending mode changes as the covalent bond character of the metal-anion

increases. It will also be important to further investigate the role of lattice-

based eigenvectors in the reactivity of these materials. As was

demonstrated for 𝛼-NaN3, the highest frequency lattice mode does permit

formation of a reactive N3∙ species. However, this pathway is unlikely to be

responsible for impact-induced initiation, given that α-NaN3 is well known

to be insensitive to this form of mechanical stimulation.

• A lattice mode in α-NaN3 was found to be sufficient to induce metallisation

at large eigenvector displacement. However, α-NaN3 is not known to be

sensitive to impact. Further insight into why this vibrational mode does not

lead to impact-induced initiation is therefore required.

• Investigating the initial stages of lattice excitation by dynamics simulations

would offer considerable insight into the up-pumping model. In particular,

understanding how the initial energy is inserted into the crystalline material,

and how this initial excitation varies as a function of material structural type.

143

• The model employed in the present contribution is based on T=0 K. Hence,

it does not offer a mechanism for understanding the temperature

dependence of impact sensitivity. The introduction of temperature by

means of thermal populations into the scattering equations would offer a

new direction for the application of this model.

• Many initiation models suggest that local defects are crucial to the process.

It would therefore be of interest to introduce computationally tractable

models for defects. Initially, this could be done by introduction of electronic

defects into the band structure calculations.

• It is clear that the dissociation energy cannot be neglected in a complete

model of initiation. Introducing a correlation term between the relative

dissociation energy of the azido anion and the quantity of up-pumped

vibrational energy at Ωmax would offer an important step forward in

generalization of this model.

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Chapter 4

Vibrational Up-Pumping in Some Molecular

Energetic Materials

4.1 Introduction

Many commonly used energetic materials (EM) are composed of organic

molecules, with well-known examples including 1,3,5,7-tetranitro-1,3,5,7-

tetrazocane (HMX), 2,4,6-trinitrotoluene (TNT) and triaminotrinitrobenzene

(TATB). These compounds are typically based on similar structural moieties

(explosophores), often -NO2 functionality, or numerous N-N, N-O or C-N bonds.

However, despite these structural similarities, organic EMs exhibit a broad

range of impact sensitivities. For example, ϵ-CL-20 (HNIW) initiates with only

ca. 3 J of impact energy,1 while TATB requires > 122 J for impact initiation.2

Particularly striking is the difference in impact sensitivity of TNT and picric acid,

whose structures differ only in a single substituent on their aromatic rings. Yet,

while TNT initiates at ca. 40 J,3 the initiation of picric acid requires

approximately half the energy, ca. 22 J.3 As organic EMs are metal-free, they

are particularly promising materials from an environmental perspective.

Consequently, there has been considerable interest in developing new organic

EMs with a broad range of energetic properties, and large libraries of such

molecules now exist. The synthesis of novel organic EMs can require very

complex and harsh experimental procedures. With no a priori knowledge of

the physical properties of the final (or indeed intermediate) products, the

process is both expensive and potentially very dangerous.

152

A number of theoretical approaches have so far been developed in an attempt

to rationalise the sensitivity of organic EMs (Chapter 1.3).4 Briefly, these have

included the study of electrostatic potentials,5,6 bond dissociation energies,7–10

large-scale empirical fitting11,12 as well as consideration of static and dynamic

band gaps.9,13,14 These methods have offered rationalisation of the impact

sensitivity of organic EMs to varying degrees. However, these models do not

typically include a physical underlying mechanism to link a mechanical impact

and the subsequent conversion of energy into a chemical reaction. Thus, the

reasons for a material displaying a particular impact sensitivity remain elusive.

The concept of vibrational up-pumping has been demonstrated experimentally

and theoretically for a range of molecular compounds.15–17 This has led to

scattered analyses of energy transfer rates in molecular EMs based on

inelastic neutron scattering spectra,18,19 or bench-top Raman spectra.20,21

While these early works did suggest strong correlation between sensitivity and

energy transfer rates, the data on which they were based was limited, and the

models inconsistent. Most notably, the mechanism of up-pumping (overtone20

or combination18,19 pathways) and the target region into which up-pumped

energy is considered, differ considerably between these models. Up-pumping

phenomena have demonstrated capable of localising energy,17,22 i.e.

producing ‘hot-spots’, and therefore offer a fundamental approach to

understanding the impact and shock-induced chemistry of EMs.17

The theoretical and experimental techniques required to enhance the

fundamental understanding of up-pumping models in solids have only recently

become available. A very recent study successfully attempted to correlate

zone-centre overtone structure with the impact sensitivity of a selection of

organic EMs.23 Having constructed a more complete picture of these

processes from first principles in Chapter 3, it is therefore of interest to further

develop and apply the model to a more challenging series of organic systems.

153

4.2 Aims

Following on from the success of the up-pumping model to predict the relative

ordering of impact sensitivity for the azide materials, this chapter aims to

extend the model to a series of molecular EMs. In doing so, the work in this

chapter seeks to:

• Consider the ‘band gap’ criterion for a series of organic EMs

• Obtain the full vibrational spectrum for a range of organic EMs

• Further develop the vibrational up-pumping mode of Chapter 3 to more

complex, large molecule materials.

• Compare and unify overtone- and combination-based up-pumping

approaches.

4.3 Model Systems

Typical organic EMs contain a large number of atoms and, common to

molecular materials, crystallise in large, low symmetry unit cells. Molecular

materials were selected to ensure calculations were computationally tractable,

and to ensure that they spanned a broad range of impact sensitivities and

structure types. Many of these materials have been studied extensively,

although a breadth of impact sensitivities are still reported for the same

compound.24 This pays testament to the difficultly in recording reproducible

impact sensitivity data. Where possible, experimental values of ℎ50 were

selected – i.e. the drop height at which 50% of tests result in initiation (Chapter

2.2.3), and a summary of common literature impact sensitivities is given in

Table 4.1.

154

Table 4.1: Experimental impact sensitivities (IS). Values of ℎ50 correspond to the height (cm) from

which a 2.5 kg weight is released. The corresponding energy is quoted as 𝐸50 (J). Sensitivities

based on unknown testing criteria are labelled as UTC. These may represent ℎ50 or the limiting

impact value.

Material Acronym IS (h50) /cm

IS (𝐸50) /J

IS (J). UTC

Ref.

Triacetonetriperoxide TATP -- -- 0.3 25 1,1’-azobistetrazole ABT -- -- < 1 26 Hexanitrobenzene HNB 11 2.75 -- 27 1,3,5,7-Tetranitro-1,3,5,7-tetrazocane

HMX 32 29 26

8 7.25 6.5

-- 1 2 2

5,5’-Hydrazinebistetrazole HBT -- -- > 30 J 28 1,1-Diamino-2,2-dinitroethene FOX-7 126 31.5 -- 29 Nitrotriazolone NTO 291 72.75 -- 2 Triaminotrinitrobenzene TATB 490 122.5 -- 2

The trend in experimental impact sensitivities adopted in this work, which will

be used to assess the output of the computational model, is based on the

following literature observations:

1. TATB is widely acknowledged to be insensitive to impact.4 Most reports

state its impact sensitivity to be ‘immeasurable’. Only a single source

suggests TATB to have a sensitivity of ca. 50 J,24 although no indication for

the origin or experimental methodology for this value was reported.

2. NTO is also generally well regarded as a highly insensitive material.24

3. Despite discrepancies in the sensitivity value (ranging from ca. 20-50 J),24

it is well accepted that α-FOX-7 is less sensitive than 𝛽-HMX.24

4. HNB is well known to be a highly sensitive compound, and more sensitive

than β-HMX.29

The trend used in this work therefore follows as ABT > HNB > HMX > HBT >

FOX-7 > NTO ≈ TATB. It is worth noting that ABT sensitivity is not quoted as

an h50 statistical value; the quoted value may thus be the limiting sensitivity

value. Due to the size of the TATP unit cell, it is considered only at the end of

this chapter. It is the most sensitive of all the materials explored here.

The unit cell and molecular structure of each material is given in Figure 4.1

155

Figure 4.1: Molecular diagrams and crystallographic unit cells for the organic EMs used in this work. The space group (SG) is shown in each case. The structure

of TATP is considered independently in Section 4.5.3.4.

156

4.4 Methods

Sample Preparation.

NTO synthesis. 3-nitro-1,2,3-triazol-5-one (NTO) was prepared by

nitration of TO (prepared by Dr. S. Kennedy, School of Chemistry,

University of Edinburgh), according to Scheme 4.1.30 TO (ca. 5 g, 60

mmol) was dissolved in excess concentrated nitric acid (70%), and held

at 55 oC for ca. 45 minutes. The nitration was quenched in an ice bath,

filtered and rinsed in ice water. Purity was verified by X-ray powder

diffraction.

Scheme 4.1: Synthetic approach for the synthesis of NTO.

Synthesis of FOX-7. Preparation of 1,1-diamino-2,2-dinitroethene

(DADNE, FOX-7) followed from Scheme 4.2.30 2-methyl-4,6-

pyrimidinedione (6.0 g, 0.05 mol) was dissolved in concentrated H2SO4

(95%, 45 mL), at temperatures < 30 oC. Concentrated HNO3 (99%, 20

mL) was subsequently added dropwise, maintaining temperatures < 20

oC, and the sample stirred for 3 hours. The resulting material was rinsed

in concentrated H2SO4, added to deionized water (100 mL) and stirred

for 2 hours. The resulting product was analysed by 1H and 13C NMR in

DMSO, and subsequently by X-ray powder diffraction.

Scheme 4.2: Synthetic approach for preparation of FOX-7.

157

TATB and 𝛽 -HMX. Samples of these materials were taken from

available stock. β-HMX was provided by the Cavendish Laboratories,

University of Cambridge. TATB was synthesised by C. Henderson

(School of Chemistry, University of Edinburgh). Samples were used as

provided, without further purification.

Condensed Matter Calculations.

The PBE-D2 scheme has been previously demonstrated to work very well for

molecular energetic materials.31–33 It was therefore chosen as a starting point

for the calculations presented here. All input structures were taken from the

Cambridge Crystallographic Database (CCDC): β-HMX (Ref: OCHTET1534),

α -FOX-7 (Ref: SEDTUQ0335), NTO (Ref: QOYJOD0636), HNB (Ref:

HNOBEN37), ABT (EWEYEL26), HBT (TIPZAU28), TATB (TATNBZ38) and

TATP (HMHOCN0739). Optimisation was performed using plane wave DFT as

implemented in CASTEP v16.40 The electronic wavefunction was expanded in

plane waves to a kinetic energy of 1800 eV for all systems, except HBT (cut-

off 1600 eV), and β-HMX, TATB and TATP (cut-off 1300 eV). Forces were

converged to < 5 × 10−4 eV/ Å, and stresses to < 5 × 10−4 GPa. The energy

change per atom was converged to < 1 × 10−9 eV/atom. The resulting unit cell

parameters are given in Table 4.2. In all cases, norm-conserving

pseudopotentials were used, as were available within the CASTEP v16

software package. All phonon calculations were based on the primitive unit

cells. Phonon calculations were performed using the linear response method

to calculate the dynamical matrices on a regular grid of wave vectors, and

Fourier interpolated to a fine grid of > 150 points for generation of phonon

density of states (DOS, 𝑔(ω)). Phonon dispersion curves were generated by

Fourier interpolation of the computed dynamical matrices along high symmetry

paths as proposed by SeeKPath41 and labelled by IUPAC convention. For

TATP, only the zone-centre phonons were calculated.

158

Electronic band structures were calculated in CRYSTAL1742 using localised

basis sets, available from the CRYSTAL17 database and selected due to

previous success with similar materials and DFT functionals (H-

H_pob_TZVP_201243; C- C_m-6-311G(d)_Heyd_200544; N- N_m-6-

311G(d)_Heyd_200544; O- O_m-6-311G(2d)_Heyd_200544. To ensure closest

reproduction of experimental results, all calculations were performed on the

experimental geometries. The band structures were calculated using the

HSE06,44 B3PW9145 and PBE46 functionals. For all materials, the tolerances

(TOLINTEG) were set at 7 7 7 9 30 (as recommended for use with these basis

sets44). The electronic structure was sampled across a regular grid of points,

with ca. 120 points sampled in each material.

Inelastic Neutron Scattering Spectroscopy. All INS spectra were collected

using the TOSCA spectrometer at the ISIS Neutron and Muon source.47,48

Samples (ca. 1.5 g) were placed in aluminium sample holders. Samples were

cooled to ca. 10 K and collected for a total of ca. 400 μAh. The sample

temperature was subsequently heated in steps of 50 K to a maximum of 200

K for β-HMX and TATB, and 150 K for α-FOX-7. Data were collected at each

50 K interval. Both forward and back-scattered data were summed and

corrected for scattering from the sample holder and background. All data

processing was done using Mantid.49 Simulated INS spectra were generated

using ABINS,50 as implemented in Mantid. Only first order quantum events (i.e.

the fundamentals) are considered in the simulation of INS spectra.

Density of States. All 𝑔(𝜔) are inherently normalized to 3𝑁. Consistent with

the ‘indirect’ up-pumping model (i.e. where up-pumped energy thermalises

across the internal vibrational manifold), the two-phonon density of states, Ω(2)

is normalized by ∫ (𝑔(𝜔)). This follows the procedure suggested previously for

the treatment of molecular materials.19,23

159

Table 4.2: Optimised unit cell parameters for the molecular EMs studied here using the PBE-D2 functional. The error in total volume is given with respect to

the experimental volume. Low temperature data are used where available. Where conventional cells are non-primitive, the primitive cell is also displayed.

a b c 𝛼 𝛽 𝛾 V dV/%

ABTexp Pbca 8.352 6.793 11.614 90 90 90 658.962

ABTcalc 8.354 6.789 11.618 90 90 90 658.931 -0.004%

HNBexp C2/c 13.220 9.130 9.680 90 95.500 90 581.490

HNBexp P(C2/c) 9.046 9.046 9.680 62.280 62.280 60.620 581.492

HNBcalc 9.028 9.028 9.771 62.444 62.444 59.981 582.977 +0.26%

𝛽-HMXexp P21/c 6.525 11.024 7.362 90 102.642 90 516.675

𝛽-HMXcalc 6.624 11.256 7.373 90 102.222 90 537.299 +4.00%

HBTexp C2/c 12.401 5.513 9.835 90 115.570 90 606.565

HBTexp P(C2/c) 6.786 6.786 9.835 113.230 113.230 47.940 303.283

HBTcalc 6.706 6.706 9.724 111.837 111.837 49.440 303.075 -0.07%

𝛼-FOX7exp P21/n 6.934 6.622 11.312 90 90.065 90 519.470

𝛼-FOX7calc 7.089 6.623 11.440 90 91.273 90 530.898 +2.20%

NTOexp P-1 5.123 10.314 17.998 106.610 97.810 90.130 902.060

NTOcalc 5.159 10.461 17.686 107.247 97.777 90.056 902.450 +0.04%

TATBexp P-1 9.010 9.028 6.812 108.580 91.820 119.970 442.524

TATB 9.128 9.142 6.767 109.012 92.097 119.936 448.784 +1.41%

TATPexp P21/c 11.964 14.038 15.595 90 117.270 90 2327.700

TATPcalc 11.913 13.811 15.415 90 117.243 90 2255.047 -3.12%

160

4.5 Results and Discussion

4.5.1 Electronic Structure

For large systems, the use of high level functionals such as HSE06 are

computationally demanding. It has been shown that the hybrid GGA functional

B3PW91 is somewhat cheaper, and offers excellent agreement with

experimental results for electronic band gap (𝐸𝑔) prediction across a broad

range of inorganic materials.51,52 It was therefore of interest to consider these

two methods for application to molecular materials, and compare to a standard

GGA functional, PBE. Only limited experimental data is available for 𝐸𝑔 for the

materials studied here. Experimental UV-Vis spectra have been documented

for 𝛽-HMX.53 While the fundamental band gap (i.e. the difference between the

ionization potential and electron affinities) is in principles different to the optical

band gap (which is stabilized by electron-hole interactions), the discrepancy is

often small in solid state materials,54 and hence the UV-Vis spectra should

offer a good indication as to the validity of the calculated 𝐸𝑔 values.

Furthermore, it should be noted that α-FOX-7 and TATB are both yellow

powders, with the former being more strongly coloured. This indicates an

optical transition in the region of ca. 2.6 eV for both materials.

The values of 𝐸𝑔 calculated for the series of molecular energetic compounds

are given in Table 4.3 (Note TATP was omitted from this part of the study as

the large unit cell renders the band structure calculation intractable). The

B3PW91 functional consistently predicts values of 𝐸𝑔 that are slightly higher

than HSE06 results, ranging from 𝐸𝑔(HSE06)+0.24 eV to 𝐸𝑔(HSE06)+0.32 eV.

Hence it appears that on average, the B3PW91 results should be within the

same approximate accuracy as the HSE06 results for related systems. As is

expected, the PBE calculations return considerably lower 𝐸𝑔 values than the

higher level functionals. Literature values for PBE-based calculations in Table

4.3 differ only slightly from those calculated here. This is most notable for β-

HMX, although literature reports are based on plane-wave basis sets (which

contrasts with the localised basis sets used in this work). In all cases, the G0W0

161

calculations found in the literature suggest a larger band gap than calculated

by either B3PW91 or HSE06. Given the limited experimental values, and

noting the colour of the TATB and FOX-7 materials, it can be inferred that G0W0

quasi-particle methods may be overestimating the values of 𝐸𝑔. This has been

demonstrated previously for inorganic systems.51

As was noted for the azide materials in Chapter 3, there is no visible trend in

the band gap values and the reported impact sensitivity of these compounds.

Based on the B3PW91 or HSE06 calculations, the predicted sensitivity

ordering would be NTO > α-FOX-7 ≈ HNB ≈ TATB > ABT > β-HMX > HBT.

This is clearly incorrect when compared to experimental sensitivity ordering.

Moreover, the agreement with the experimental ordering worsens if the values

from G0W0 are considered. As was also observed for the azide materials, there

is no evidence of any correlation between sensitivity and a material having a

direct or indirect band gap.

Table 4.3: Fundamental electronic band gaps (𝐸𝑔) in the crystalline molecular energetic materials,

arranged in order of decreasing impact sensitivity. All values calculated here are based on a

localised basis set. 𝐸𝑔 are labelled as direct (D) or indirect (I) band gaps.

Material B3PW91 PBE HSE06 Lit. Calc Lit. Exp.

ABT 5.0317 (I) 2.9982 (I) 4.7882 (I) -- --

HNB 3.9433 (I) 2.1040 (I) 3.6887 (D) -- --

𝛽-HMX 5.4954 (D) 3.6826 (I) 5.2176 (D) 7.21,҂ a 4.66 ^a 5.32*

HBT 5.9569 (I) 4.2069 (I) 5.6364 (I) -- --

𝛼-FOX-7 3.9833 (I) 2.4483 (I) 3.6719 (I) 5.1,҂,b 2.2,^,c 1.9^,b --

NTO 3.5024 (I) 2.1027 (D) 3.1892 (I) -- --

TATB 3.9824 (I) 2.6334 (I) 3.6599 (I) 4.66,҂ a 2.52^a --

҂ G0W0 from (a) Ref 55 (b) Ref 56; ^ PBE from (a) Ref 55 (b) Ref 57 (c) Ref 58; * From Ref 53 based

on UV-Vis spectroscopy.

4.5.2 Vibrational Structure of Some Organic Energetic Materials

Following from the theory presented in Chapter 3, the vibrational structure of

the molecular materials was considered. The full phonon dispersion curves

162

were calculated along the high symmetry lines of the Brillouin zones and are

given in Figure 4.2. As expected for molecular materials, it is generally seen

that the branch dispersion is relatively small across the Brillouin zone, and

almost negligible for the internal vibrational modes.

Slight instabilities are observed in the phonon dispersion curves of both NTO

and HNB, with frequencies of the lowest acoustic branch becoming negative

at a small set of wave vectors. Unfortunately, no attempts to rectify this were

successful. As both compounds are stable, this is unlikely to be indicative of

dynamic instability of the structures, but rather more likely to be attributed to a

slight numerical error in the calculated structures. However, this is not

expected to result in any marked effect on the remaining vibrational structure.

As a means to assess the ability of DFT to model the vibrational structure for

these types of compounds, experimental INS spectra for a subset of the test

compounds were collected. Calculated phonon dispersion curves were then

used to simulated INS spectra for direct comparison.

The simulated INS spectra for the most sensitive compound, β-HMX, Figure

4.3, generally shows good agreement with experiment. The frequencies of the

lowest region of the INS spectrum are well reproduced, with the calculated

Ω𝑚𝑎𝑥 underestimated by only ca. 5 cm-1. There appears to be a ca. 20 cm-1

systematic underestimation of the vibrational frequencies in the ω > 200 cm-1

region of the spectrum. The intensities are not well reproduced for the lowest

frequency modes, which suggests some error surrounding the exact structure

of the β-HMX phonon modes, or textured powder. In contrast, comparison of

the simulated and experimental INS spectra for α-FOX-7, Figure 4.3, shows

excellent agreement. While the frequencies are well reproduced based on the

zone-centre structure, increased sampling of the Brillouin zone is required to

obtain accurate intensities. By 0.08 Å-1, the full INS spectrum is very well

reproduced by simulation.

163

Figure 4.2: Phonon dispersion curves for the molecular EMs discussed in this work. Wavenumber is truncated at 600 cm-1 to allow visualisation of the low

frequency modes that are important in this work.

164

Figure 4.3: Inelastic neutron scattering spectra for β-HMX and α-FOX-7 at 10 K. The experimental

pattern (black) is given in comparison to (blue) simulated INS spectra at three different sampling

densities of the Brillouin zone. Only the first quantum events are simulated.

For NTO the simulations are an excellent reproduction of the experimental INS

spectra, Figure 4.4, despite the small instability (negative frequencies)

reported above. Both the intensities and frequencies appear well reproduced

across the spectrum, particularly when a dense grid of phonon wave vectors

165

is used. This strongly suggests that this minor instability has negligible

influence on the vibrational structure. The INS spectrum of TATB is also well

reproduced, Figure 4.4, but there does appear to be a slight disagreement

between the higher frequency vibrational modes at ca. 800 cm-1. Internal

modes in this region are primarily NH2 twisting modes. Overall, however, the

spectrum is very well reproduced in both frequencies and intensities.

In contrast to the case of α -NaN3 presented in Chapter 3, INS spectra

simulated from zone-centre calculations generally perform well in reproducing

both the frequencies and relative intensities observed in the experimental INS

spectra of the organic molecular materials. The notable exceptions to this are

the lowest frequency lattice modes, which appear to converge to the

experimental spectra when the Brillouin zone is sampled at ca. 0.04 Å-1.

Increasing the density of q-point sampling for simulation of the INS spectra has

negligible effect on frequencies above ca. 200 cm-1. These features are

presumably due to both the minimal dispersion observed across the Brillouin

zone (Figure 4.2), as well as the dominant incoherent scattering of the

hydrogen in these materials. Overall, it does appear that the vibrational

structure of these molecular materials is well reproduced by the simulations,

particularly in the low frequency regions. DFT therefore appears capable of

producing an accurate description of not only the zone-centre frequencies, but

of the dispersion relationship through the Brillouin zone for these types of

materials. The vibrational structures used herein can therefore be taken as

representative of the true materials.

166

Figure 4.4: Inelastic neutron scattering spectra for NTO and TATB at 10 K. The experimental

pattern (black) is given, in comparison to (blue) simulated INS spectra at three different sampling

densities of the Brillouin zone. Only the first quantum events are simulated.

167

4.5.3 Vibrational Up-Pumping in the Molecular Energetic Materials

Prior to considering the vibrational up-pumping in the molecular EMs, it is again

necessary to segment the vibrational spectra into sections based on integer

values of Ω𝑚𝑎𝑥. The definition of Ω𝑚𝑎𝑥 stated in Chapter 3 is somewhat less

clearly defined when considering the molecular materials. A good example of

this is NTO. Across the phonon density of states (DOS, 𝑔(ω)) there are clear

minima near the top of the phonon region which, when compared to the

phonon dispersion curves in Section 4.5.2, do correlate to regions of gaps,

albeit small, in phonon density. The non-zero values of 𝑔(ω) result from the

applied Gaussian broadening on generation of the DOS. Previous works have

suggested that in such cases, the top of the phonon bath should be taken to

include (nearly) amalgamated NO2 rocking modes,19,20,23 and therefore act as

the upper limit of the phonon region. For NO2 containing compounds, these

modes are shown in Table 4.4. However, the DOS and phonon dispersion

bands clearly indicate a frequency gap between the top of a semi-continuum

and the highest -NO2 rocking modes at ca. 230 cm-1. These highest rocking

modes therefore do not fit within the continuum criteria for defining the phonon

bath. Where appropriate, consideration is given for Ω𝑚𝑎𝑥 placed in both

locations for NTO. Note that the potential Ω𝑚𝑎𝑥 at 170 cm-1 indicated by the

INS spectra is not considered further as it is lost due to the addition of

Gaussian broadening in the calculated spectra, which is added to reflect

resonant states.20,23 This leads to the placement of Ω𝑚𝑎𝑥 as highlighted in

Figure 4.5 and reported in Table 4.4.

168

Figure 4.5: Vibrational density of states (𝑔(ω)) for the molecular energetic materials arranged in

order of decreasing impact sensitivity. The segments of integer values of Ω𝑚𝑎𝑥 are indicated in

each case.

169

Table 4.4: Vibrational structure (cm-1) of the molecular energetic compounds. The top of the

phonon bath Ω𝑚𝑎𝑥, first doorway mode ω𝑑, and frequency gap (Δω𝑑 = ω𝑑 − Ω𝑚𝑎𝑥).

NO2 rock max

𝛀𝒎𝒂𝒙 (INS)

𝛀𝒎𝒂𝒙 (CALC)

𝝎𝒅 𝚫𝝎𝒅

ABT -- -- 175 220 45

HNB 200 -- 210 245 35

𝛽-HMX 166 195 193 210 15

HBT -- -- 200 225 25

𝛼-FOX-7 155 183 185 255 70

NTO 240 170,202,245 200/240 220 / 325 20/85

TATB 155 155 160 234 74

The decomposition pathways of molecular energetic materials are complex59,60

and remain largely unknown. Compared to the structurally simpler azide

compounds discussed in Chapter 3, it is highly probable that many normal

modes are simultaneously required to initiate the decomposition of these large

molecules. It is therefore unlikely that a direct up-pumping mechanism (i.e.

energy localisation into a single vibration, and immediate bond rupture) occurs.

Rather, it is more likely that an indirect (or thermal)61 mechanism occurs,

whereby the excited molecule reacts at some point following evolution of the

highly excited vibrational state.17 The redistribution of vibrational energy within

the internal molecular vibrational manifold is relatively quick, and once energy

reaches this manifold it can quickly redistribute. In contrast to the azide

systems, the rate determining step is instead taken to be the transfer of energy

from the phonon manifold into the internal vibrational manifold.20,23 This occurs

in two steps: (1) population of the doorway modes (i.e. modes with Ω𝑚𝑎𝑥 <

ω < 2Ω𝑚𝑎𝑥) and (2) population of higher-lying modes.61

The concept of the ‘target’ frequency, ω𝑇, is therefore no longer meaningful,

as there is not a single (or known subset) of vibrational modes into which shock

wave energy must be localised to induce a chemical response. Hence, it is no

longer appropriate to consider the ‘frequency gap’ criterion based on Δω =

ω𝑇 − Ω𝑚𝑎𝑥. Instead, this simple criterion can be recast as Δω = ω𝑑 − Ω𝑚𝑎𝑥,

where ω𝑑 is the first doorway mode. If this assessment is made based on the

170

full phonon dispersion curves, the resulting sensitivity ordering follows that

shown in Figure 4.6. If Ω𝑚𝑎𝑥 for NTO is taken to be 238 cm-1 (i.e. the top of the

-NO2 rocking modes) only a very weak correlation is observed. The more

sensitive compounds tend to have smaller values of Δω than the less sensitive

materials. This trend is more apparent if the energetic materials with

explosophoric -NO2 moieties are instead considered in isolation, and may

therefore suggest additional electronic factors may be important in determining

sensitivity ordering (red symbols in Figure 4.6). However, while this correlation

may be indicative from an initial screening perspective, it is greatly limited

beyond a very rough energy classification perspective.

Figure 4.6: Predicted sensitivity order based on the vibrational frequency ‘energy gap’ criterion,

distinguishing between compounds containing -NO2 groups (red squares) and those that do not

(black squares).

Noting that the up-pumping model relies initially on the rate of excitation of the

doorway frequencies, an alternative qualitative correlation can be sought

between the doorway density and impact sensitivity. If this is instead

considered, there does appear to be a good overall correlation with sensitivity,

171

Figure 4.7. The more sensitive compounds contain a higher density of doorway

modes in the range Ω𝑚𝑎𝑥 → 2Ω𝑚𝑎𝑥 , with the less sensitive compounds

exhibiting lower densities of states within this region. The notable exception to

this rule is ABT. The chemical structure of ABT is considerably different from

the remaining compounds studied here, and again it can be suggested that

electronic effects dominate in dictating the different sensitivity of this

compound.

While this method does not offer high resolution of the sensitivity ordering (that

is, that α-FOX-7 is predicted to be more sensitive than β-HMX), it does offer a

relatively rapid, qualitative approach to the general classification of these

materials once vibrational frequencies have been obtained (by calculation or

experimental means).

Figure 4.7: Comparison of doorway density of states and experimental impact sensitivities.

Doorway densities are normalized by 3N to account for variations in the normalization of the DOS.

While the qualitative trends suggested above do display some promise in their

ordering of the impact sensitivity of these molecular compounds, they do not

offer much by means of a physical mechanism. Hence, it is worthwhile

172

returning to discussion of the up-pumping methodology employed in Chapter

3.

The major difference in the present case, compared to the azide series, is the

lack of a well-defined target frequency, ω𝑇 . For the present, ignoring any

explicit consideration of temperature, the model in this section makes the

following assumptions:

1. Overtone pathways are responsible for the initial transfer of energy.61

Energy transfer via the first overtone is considerably faster than by

higher order overtones, and hence the region up to 2Ω𝑚𝑎𝑥 quickly

becomes populated. This leads to the definition of the doorway modes

as having frequencies, Ω𝑚𝑎𝑥 < ω < 2Ωmax.

2. Initial energy transfer that results from overtone up-pumping into the

doorway modes can subsequently up-pump via combination pathways.

3. It follows from (2) that combination pathways are limited to the excitation

of modes below a maximum of 3Ω𝑚𝑎𝑥 . That is to say that mode

combinations can further populate other, higher frequency doorway

modes, or they can populate higher frequency internal modes.

Secondary combination pathways, including vibrational cooling, are not

considered.

4. The energy up-pumping model is based on the total number of available

pathways.

5. Overtone pathways lead to initial excitation of the vibrational manifold

to a maximum of 2Ω𝑚𝑎𝑥 from the first set of overtones. The second

overtone can excite to a maximum of 3Ω𝑚𝑎𝑥 albeit at a slower rate.

Higher order processes may occur at even lower rates, but are not

competitive. This is due to the rapidly decreasing probability of higher

order scattering events.22

6. The population of the phonon bath is assumed to remain constant, and

the contributions of overtones and combinations are taken as being fully

separable: i.e. they do not compete. This holds approximately for the

initial energy transfer step.61

173

Hence, the total energy transfer into the molecular vibrational region is again

dominated by the fastest combination and overtone processes.

Due to the markedly different molecular and crystallographic structures of

these materials, the two phonon density of states is recast as:

Ω(2) = ρ(ω)−1 ∫𝑑ωδ(ω1 − ω2 − ω3)

Equation 4.1

where ρ(ω) is the total density of states. This has the effect of normalising the

up-pumping contribution by 3𝑁 and reflects the dissipation of up-pumped

energy into the internal vibrational manifold. Furthermore, noting that up-

pumped energy is only meaningful if a real vibrational state exists at the

resulting energy, Ω(2) is projected onto ρ(ω), generating the projected two-

phonon density of states, P(Ω(2)). As the latter was generated with a Gaussian

broadening of 10 cm-1, this process accounts for potential resonance

pathways.20,23

4.5.3.1 Overtone Pathways

It has been previously suggested that overtone pathways are sufficient to

model the relative up-pumping rates in molecular energetic compounds.20,23

Most recently, based solely on zone-centre vibrational frequencies (i.e. not

accounting for the varying density of states across the Brillouin zone),

Bernstein suggested that Ω𝑚𝑎𝑥 should be placed at 200 cm-1 for all molecular

compounds, and up-pumping into the region 200-700 cm-1 should be

considered. Earlier suggestions have imposed the restriction of Ω𝑚𝑎𝑥 at 250

cm-1.20 The suggestion of a 700 cm-1 cap appears to have propagated from

early experimental work (which placed a limit at 600 cm-1).19 However, this

experimental work employed this limit artificially, as it was the upper limit of

experimental resolution at the time. No alternative explanation has yet been

provided for this upper boundary limit. Within the framework of the model

proposed in Chapter 3, these previous works exhibit two major flaws:

174

1. Table 4.4 shows that the limiting value of the phonon bath cannot

always be taken to be 200 cm-1. This is particularly notable in cases

such as TATB, with a well-defined Ωmax at 160 cm-1.

2. There is no physical rationale for limiting up-pumping to 700 cm-1,

particularly when higher order overtones are considered, as has

previously been done.23 Given the conservation of energy, the

maximum overtone contribution from up-pumping should scale as

𝑁Ω𝑚𝑎𝑥, where N is the order of the overtone.

At low overtone numbers (N) (see Equation 3.7) the arbitrary maximum of 700

cm-1 is in fact meaningless. For example, consider the two dominant overtone

pathways, N=2 and N=3. For a system with Ω𝑚𝑎𝑥 = 200 cm-1 the maximum

overtone frequency is 400 (for N=2) or 600 cm-1 for (N=3). Hence no density

will exist above 𝑁Ω𝑚𝑎𝑥 in these cases. Despite these deficiencies, if the criteria

set out by Bernstein are followed (Ω𝑚𝑎𝑥 = 200 cm-1 overtone vibrational up-

pumping and projection onto the 200-700 cm-1 region), remarkable correlation

is made against experimental impact sensitivity, Figure 4.8. If only the first

overtone is considered (i.e. the most rapid excitation) there is a seemingly

exponential fit between the integrated overtone contributions to P(Ω(2)) as a

function of the proposed experimental impact sensitivity. This is to say that

𝑃(Ω(2)) is higher for more sensitive compounds. This is in particularly excellent

agreement if the materials which contain explosophoric -NO2 groups are

considered in isolation (red points on Figure 4.8). As N is increased, this

correlation holds quite well, with the notable exception of α-FOX-7, Figure 4.8.

When the second overtone contributions are considered (N=3), the total P(Ω(2))

for α-FOX-7 and β-HMX become nearly equal, with the latter appearing to fall

short of the exponential trend. However, consideration of Figure 4.5 for β-HMX

(which agrees well with INS data) shows that very little density sits within the

region ω = 2Ω𝑚𝑎𝑥 → 3Ω𝑚𝑎𝑥. It is therefore not surprising to find that the value

of P(Ω(2)) falls as N increases.

175

Figure 4.8: Overtone-based prediction of impact sensitivity of molecular energetic materials,

P(Ω(2)). Data are given for (left) the first overtone, N=2, and (right) the second overtone, N=2+3.

Molecules which contain explosophoric -NO2 moieties are highlighted in red, those without in

black. Up-pumping is considered into the region 200-700 cm-1 with Ω𝑚𝑎𝑥 =200 cm-1.

It is worth considering the other two compounds, ABT and HBT. Both

compounds are based on the same tetrazole base, and do not contain -NO2

moieties. Instead, their explosophoric groups are based on N-N bonds.

Interestingly, while the values of Ω(2) do not fall in line with the trend exhibited

by the -NO2 containing compounds, ABT is predicted more sensitive than HBT,

consistent with experimental reports. The offset between the two sets of

materials presumably reflects the nature of the electronic contribution to

dissociation (i.e. the dissociation energies).

While it is important to note that higher order overtones are unlikely to

contribute to up-pumping due to their lower probabilities, P( Ω(2) ) was

generated up to N=6, at which point additional contributions became negligible.

Overall, the general trend in predicted sensitivity of the molecular materials

holds well as P(Ω(2)) is generated from higher overtones, Figure 4.9. Again,

the -NO2 based materials (with the exception of β-HMX) follow a good trend in

176

Ω(2) as compared to impact sensitivity. The second structural class of

energetic materials, based on N-N explosophores, follow their own trend, but

are still in line with experiment.

Figure 4.9: Integration of Ω(2) generated from overtone pathways from N=2-6. Materials with

explosophoric -NO2 moieties are highlighted red, those without in black. Up-pumping is

considered into the region 200-700 cm-1 with Ω𝑚𝑎𝑥 =200 cm-1.

As a final consideration for the overtone pathways, the limits placed on Ω𝑚𝑎𝑥

and the upper end of integration were lifted. The upper limit of integration

therefore sits at 𝑁Ω𝑚𝑎𝑥. Upon lifting this restriction, the correlation between

impact sensitivity and P(Ω(2)) holds until a value of N=4, after which point the

sensitivity of α-FOX-7 surpasses that of β-HMX, and eventually that of HNB by

N=6. Hence, this is purely an effect of high order overtones and the result of a

markedly higher DOS in the high frequency region of α-FOX-7, (see the DOS

177

in Figure 4.5). However, it is worth remembering that these scattering

processes are highly improbable.

It can therefore be suggested that the seemingly arbitrary upper limit of

integration previously suggested (700 cm-1) was in fact a fortunate choice. This

limit effectively places the restriction on overtones to N=3, with higher order

terms contributing a negligible amount, should they be considered (as in the

case by Bernstein23). If only the first two overtone pathways are considered

(as in Chapter 3), and Ω𝑚𝑎𝑥 defined as in Table 4.4, the predicted ordering

follows as in Figure 4.10. This is in excellent agreement with experimental

sensitivities, noting the two independent sets of materials: -NO2 (red squares)

and N-N (black squares) based.

Figure 4.10: Integration of Ω(2) generated from the overtone pathways for N=2-3. The -NO2

containing materials are highlighted in red, those without in black. No restrictions are placed on

Ω𝑚𝑎𝑥 or the upper frequency bound for the doorway modes.

178

4.5.3.2 Combination Pathways

The initial up-pumping models62 included consideration of combination

pathways, which later formed the base for prediction of impact sensitivity from

INS spectra.19 In the 0 K limit, combination pathways cannot contribute until

doorway modes have been populated by overtone processes. That said, Kim

and Dlott61 noted that the initial overpopulation of doorway modes (by overtone

pathways) is small, and that the subsequent excitation of higher vibrational

modes (i.e. by combination pathways) follows quickly afterwards (1-2 ps).

Hence, it is worth analysing the contributions of combination pathways and

their potential to rationalise impact sensitivity.

The combination pathways generated in the absence of temperature are

simply taken as the two-phonon density of state, Ω(2) = δ(ω − ω1 − ω2), with

ω1 ≠ ω2, in line with Equation 3.5. As discussed in Chapter 3, a restriction is

placed on the generation of these curves, such that ω1 < 2Ω𝑚𝑎𝑥 and ω2 <

Ω𝑚𝑎𝑥. Hence, the maximum allowed target frequency is 3Ω𝑚𝑎𝑥. This has the

effect of ensuring that up-pumping occurs by the addition of at least one mode

from the phonon bath, which is initially excited by the impact of the shock wave.

These Ω(2) curves are generated for the materials studied here, Figure 4.11.

As a qualitative rule, Ω(2) appears to increase earlier and more rapidly for the

sensitive compounds of each structure type. For example, the onset of

increase is roughly the same between HNB (~ 260 cm-1) and β-HMX (~ 250

cm-1), although the former rises much more rapidly. In contrast, α-FOX-7 has

an onset frequency (~ 320 cm-1) approximately 100 cm-1 higher than in β-HMX.

In general, each successive doorway mode leads to an increase in Ω(2) that

corresponds to the density of states about that doorway mode. Hence, in line

with Fermi’s Golden Rule, Equation 3.6, a lower onset frequency and more

rapid increase in Ω(2) corresponds to a more rapid transfer of energy into the

internal modes. A notable exception to this generalization appears to be NTO,

which is based on Ω𝑚𝑎𝑥 = 200 cm-1 in Figure 4.11. The rapid onset of Ω(2)

results from the low-lying vibrational band that sits just above Ω𝑚𝑎𝑥. If NTO is

instead recast based on Ω𝑚𝑎𝑥=240 cm-1, the onset frequency is shifted to

179

~350 cm-1, Figure 4.11. This further supports previous suggestions to include

the -NO2 rocking motions within the phonon bath.

Figure 4.11: Ω(2) for the molecular energetic materials. Ω(2) for NTO is given for (black) Ω𝑚𝑎𝑥(𝑁𝑇𝑂)

=

200 cm-1 and (green) Ω𝑚𝑎𝑥(𝑁𝑇𝑂)

= 240 cm-1. Plots are generated according to Equation 4.1 and

under restriction of ω1 < 2Ω𝑚𝑎𝑥 and ω2 < Ω𝑚𝑎𝑥. Vertical dotted lines indicate Ω𝑚𝑎𝑥.

180

Raw integration of Ω(2) results in largely meaningless quantities, noting that

up-pumped energy again only contributes to the excitation of the internal

modes if an internal mode exists at a particular frequency. Hence, Ω(2) are

again projected onto the DOS curves, and P(Ω(2)) are generated. An example

is given for ABT in Figure 4.12. As is observed for ABT, and indeed holds

across the energetic materials, the large majority of Ω(2) sits between existing

vibrational states and can therefore be discarded.

Figure 4.12: The full Ω(2) for ABT (black), alongside the single phonon DOS g(ω) (orange) and the

projection of Ω(2) onto the DOS, P(Ω(2)) (blue).

With these preparations in mind, it is possible to analyse the combination mode

contributions to the energy transfer of the molecular EMs within the 0 K model.

In the first instance, this simply corresponds to an integration of the P(Ω(2))

curves generated above.

181

The integration of P(Ω(2)), Figure 4.13, does not reveal as promising a trend

as was observed for the azide materials in Chapter 3. There is no exponential

decay observed with increasing impact sensitivity, and the integrated values

of α-FOX-7 and β-HMX are very similar. In fact, α-FOX-7 is predicted to be

slightly more sensitive than β-HMX. This may point towards an error with the

assignment of Ω𝑚𝑎𝑥 in the computational model. However, inspection of the

INS spectra suggests that the values employed in both cases are accurate,

and that no peaks are added or omitted to the region Ω𝑚𝑎𝑥 → 3Ω𝑚𝑎𝑥 in the

simulated spectra for either compound. These therefore appear to be well

representative of the systems under the current model.

Figure 4.13: Integration of Ω(2) from combination pathways. Compounds containing -NO2

explosophores are highlighted in red. Note that Ω(2) is restricted to a maximum of 3Ω𝑚𝑎𝑥 given

the restrictions of ω1 < 2Ω𝑚𝑎𝑥 and ω2 < Ω𝑚𝑎𝑥.

It is generally observed that P(Ω(2)) is higher for the sensitive compounds and

lower for the insensitive compounds, but the resolution is very poor. The failure

of this model is likely due to the complexity of the vibrational structure of the

182

molecular materials. The number of energy transfer processes that are

available within these materials is dependent on the number and density of

doorway modes. However, the frequencies of doorway modes differ quite

drastically within and between materials, and the present model treats coupling

with all of these modes as being equal. While the anharmonic coupling

constants may be very similar,21 the number of scattering pathways available

will depend on their relative populations. Hence, doorway modes that sit higher

in frequency will contribute fewer pathways if thermal populations are

considered. To a large extend, this may explain the inability of these 0 K

models to differentiate between β -HMX and α -FOX-7. In the former, the

doorway modes tend towards the bottom of the doorway region, while in the

latter they tend towards the top. Thus, while a simple counting method

appeared sufficient to describe the vibrational up-pumping in the vibrationally

‘simple’ azide molecules, it appears inadequate to treat the more complex

vibrational structure here. Thus P(Ω(2)) alone does not appear sufficient.

4.5.3.3 Two-Layer Combination Pathways

As a first step to develop this model further within the 0 K limit, and attempting

to unify previous works, energy transfer is instead explicitly treated as the two-

step process that was described above61, namely:

1. Excitation of the doorway modes by the first overtone, followed by

2. Up-pumping by combination pathways to a maximum of 3Ω𝑚𝑎𝑥

This is done by imposing the populations of the doorway modes that result

from the overtone up-pumping calculations in Section 4.5.3 onto 𝑔(ω), and

subsequently assessing P(Ω(2)) as before. This is demonstrated in Figure 4.14

for -FOX-7.

183

Figure 4.14: Construction of the two-layered approach for vibrational up-pumping for -FOX-7.

(Top) The phonon density of states (blue) is shown, along with the first overtone density of states

(green). (Bottom) Recasting the overtone populations onto the phonon density of states in the

region Ω𝑚𝑎𝑥 < ω < 2Ω𝑚𝑎𝑥 (highlighted by red box).

If the model (bottom panel, Figure 4.14) is constructed, and the up-pumping

contributions re-examined, the predicted trend in sensitivities sits in excellent

agreement with experimental results, Figure 4.15. It is assumed here that

excitation of all modes between Ω𝑚𝑎𝑥 → 3Ω𝑚𝑎𝑥 (i.e the entire internal

vibrational manifold) should be considered. The predicted sensitivity ordering

184

follows as HNB > β-HMX > α-FOX-7 > NTO > TATB across the -NO2 based

energetics, and ABT > HBT for the N-N energetic materials. While the model

imposed here is slightly more complex than that required in Chapter 3, it does

highlight the need for a more physical basis in understanding the properties of

energetic materials with large quantities of doorway modes and complex

vibrational structure.

Figure 4.15: Relative up-pumping rates according to the two-layered model. Note that Ω(2) is

restricted to a maximum of 3Ω𝑚𝑎𝑥 given the restrictions of ω1 < 2Ω𝑚𝑎𝑥 and ω2 < Ω𝑚𝑎𝑥 .

Compounds containing -NO2 explosophores are highlighted in red.

Whilst the ordering proposed in Figure 4.15 shows excellent agreement with

experimental impact sensitivities it has been postulated61,62 that for some

materials the main target modes (e.g. bond stretching) are confined to the

region 2Ω𝑚𝑎𝑥 < ω < 3Ω𝑚𝑎𝑥 . Without a deeper understanding of the

dissociation mechanisms of these energetic materials, it is not possible to say

explicitly whether the range Ω𝑚𝑎𝑥 < ω < 3Ω𝑚𝑎𝑥 or 2Ω𝑚𝑎𝑥 < ω < 3Ω𝑚𝑎𝑥 should

185

be considered. However, it is worth highlighting that if the integration from

Figure 4.15 is restricted to the upper range, Figure 4.16 is the result. This leads

to truly excellent agreement with experimental impact sensitivity ordering,

including the positioning of ABT. Now only HBT remains as an outlier. Further

information as to which bonds require activation is therefore of great

importance in developing this model further.

Figure 4.16: Relative up-pumping rates according to the two-layered model. Only the up-pumping

contribution to 2Ω𝑚𝑎𝑥 < ω < 3Ω𝑚𝑎𝑥 is considered here.

4.5.3.4 Temperature Dependent Up-Pumping

It follows from Section 4.5.3.2 that simple counting of the number of up-

pumping pathways is not the best indicator to describe the relative sensitivities

of the molecular energetic materials. Instead, it is likely that an understanding

of the rate of this up-pumping may be more indicative. This will be examined

here within the purview of temperature.

186

It is well known that temperature can have a marked impact on the sensitivities

of energetic materials.63 With variation over relatively small temperature

ranges, two mechanisms can be proposed for the effect of temperature on the

models presented in this thesis:

1. Induce a change in the phonon bath populations, and

2. Induce a large anisotropic shift in vibrational frequencies

Temperature Effects: Phonon Bath Populations

In the absence of a thorough understanding of the three-phonon scattering

probabilities for an arbitrary set of phonons, Dlott22 noted that the relative rates

of energy up-pumping varies with:

𝑟𝑎𝑡𝑒 ∝ 𝑛𝑝 − 𝑛𝑡

Equation 4.2

That is, it decreases as the difference between the Bose-Einstein populations

of the lower (phonon, 𝑛𝑝) and upper (target, 𝑛𝑡) frequencies narrows. This is

analogous to describing the ‘heat flow’ associated with phonon up-pumping

from a vibrationally ‘hot’ phonon continuum to a vibrationally ‘cold’ internal

vibrational manifold – the closer in ‘temperature’ the initial and final states, the

slower the energy transfer.

With the addition of temperature, a two-stage model is no longer explicitly

required. The initial up-pumping of vibrational energy follows the quickest

routes, which are presumably the first overtone and combination pathways.

These both occur within the first anharmonic approximation. With addition of

temperature:

1. The initial contribution from the doorway modes no longer requires

population from the overtone pathway, as it rises due to thermally

populated states.

187

2. Combination pathways therefore contribute to scattering across ω > 2Ω𝑚𝑎𝑥.

At least one of the coupling modes must have ω < Ωmax – i.e. must

incorporate the shock temperature (the remaining system is at equilibrium).

3. The rate of up-pumping from the overtone pathways (𝜅𝑂𝑇) is determined

according to61

𝜅𝑂𝑇 = 𝐴Ω(2)[𝑛𝑝(𝑇) − 𝑛𝜔(𝑇)]

Equation 4.3

where A contains a series of scaling constants as well as the anharmonic

coupling constant 𝑉(3), Ω(2) is the two-phonon density of states, and 𝑛𝑝(𝑇)

and 𝑛𝜔(𝑇) are the Bose-Einstein populations of the phonon and target

frequencies, respectively. The coefficient A has been suggested to depend

on heat capacity and the rate of acoustic propagation in a material.

However, as this information is not available for the majority of these

materials, this term is assumed to remain constant for all systems.

4. The relative rate of up-pumping from combination pathways (𝜅𝐶) is taken

to follow19

𝜅𝐶 = 𝐴 Ω(2)[𝑛𝑝(𝑇)𝑛𝑑(𝑇) − 𝑛𝜔(𝑇)]

Equation 4.4

with terms defined as above and the addition of 𝑛𝑑(𝑇), the Bose-Einstein

population of the doorway mode. This has the effect of scaling the

magnitude of up-pumping contributions according to the thermally excited

populations of the doorway modes, Figure 4.17.

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Figure 4.17: Bose-Einstein populations as a function of frequency at (blue) 300 K, (red) 500 K and

(green) 1000 K.

Because combination bands can now result from thermally populated states,

it is first worth considering the full Ω(2) curves that are obtained on lifting the

restriction of ω1 < 2Ω𝑚𝑎𝑥 (whilst maintaining ω2 < Ω𝑚𝑎𝑥 ), Figure 4.18. This

therefore allows the excited phonon bath to interact with any thermally-

populated vibrational mode. In contrast to the phonon density of states, Ω(2)

rarely falls to zero, and only does so when neighbouring frequencies have

Δω > Ω𝑚𝑎𝑥. Thus, above such regions, rapid redistribution of energy cannot

occur within the first anharmonic approximation, and these frequencies can be

largely eliminated from a thermal (non-direct) up-pumping model.22 This is

because the thermal up-pumping model requires up-pumped energy to

dynamically redistribute into vibrational modes that are responsible for

assisting in bond rupture. Therefore the up-pumping is either limited by this

point in Ω(2) or intrinsically by the highest vibrational frequency (via generation

of P( Ω(2) ). Across the Ω(2) for these compounds, none of the sensitive

compounds contain regions of Ω(2) = 0. It is worth highlighting that α-FOX-7

does have such a point at ca. 1000 cm-1, which corresponds to the large

189

frequency gap observed in the density of states, Figure 4.5. This feature is

promising for segregating the sensitive and insensitive materials. For those

compounds that do contain Ω(2) = 0, their values are listed in Table 4.5. These

values are important as they set the upper limit for vibrational energy transfer

within the first anharmonic approximation.

Table 4.5: Limiting frequencies for the full Ω(2) for the molecular energetic materials.

𝛀(𝟐) = 𝟎 /cm-1

ABT --

HNB --

βHMX --

HBT 600

αFOX-7 1034

NTO --

TATB 1028

190

Figure 4.18: Complete temperature independent Ω(2) for the molecular energetic materials,

generated under the single restriction of ω2 < Ω𝑚𝑎𝑥. For NTO, the curves are shown for Ω𝑚𝑎𝑥 of

(green) 240 and (black) 200 cm-1. It is encouraging to find that apart from the onset frequency,

little changes as a function of Ω𝑚𝑎𝑥.

191

It is convenient to begin discussion based only on the combination pathways.

This is the result of integrating 𝑃(Ω(2)), with Ω(2) generated from a thermally

populated 𝑔(ω). If the system is first set at equilibrium temperature, T = 300 K,

the predicted order of sensitivities, Figure 4.19, are generally consistent with

the pure combination pathway prediction. Note that the absolute values plotted

on the Y-axis have markedly increased compared to the earlier T = 0 K models.

Thus, only the trends can be compared. However, the resolution between

sensitivities of β-HMX and α-FOX-7 improves. Again, the two material types

are separated, with up-pumping in the -NO2 based materials being higher and

exhibiting an exponential trend. NTO is unfortunately grossly overestimated

using this simple method. Numerically, this results from the high phonon

density of states of the NTO phonon bath, as compared to the other materials.

This is logical as it reflects the higher heat capacity of the phonon bath, given

the larger number of molecules in the primitive cell.

Figure 4.19: Predicted sensitivity based on T = 300 K, based purely on combination pathways (i.e.

integration of 𝑃(Ω(2))). Ω(2) are generated under the restriction ω2 < Ω𝑚𝑎𝑥 and integration

restricted by where Ω(2) = 0.

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Further development of the model is clearly required to resolve this situation.

In the first instance, three physical suggestions can be proposed to account

for this:

1. The thermal expansion of the material leads to softening of the phonon

bath modes, and hence a decrease in Ω𝑚𝑎𝑥

2. The bond dissociation energy of NTO is considerably higher than for

the more sensitive compounds, and electronic effects become dominant

in this compound.

3. A considerably different anharmonic constant occurs for this compound

as compared with the others in the test set.

While the two-level model may not explicitly be required, it has been observed

that up-pumping rates of overtones greatly exceed those of combination bands,

by an order of magnitude in some materials.61 Hence it is worth re-examining

the temperature effects under the construction of the two-level model of

Section 4.5.3.3 (i.e. population of doorway states by the first overtone, followed

by combination mode up-pumping). This is accomplished by first thermally

exciting 𝑔(ω) and projecting the first overtone (i.e. N=2) onto the doorway

frequencies as in Figure 4.14. The remaining 𝑔(ω) remains thermally

populated, and the combination pathways (i.e. Ω(2) ) are calculated,

maintaining ω2 < Ω𝑚𝑎𝑥 . For the purpose of this discussion, the equilibrium

temperature is set at 300 K. Compared to the model built upon thermally

populated combination bands alone (i.e. Figure 4.19), as well as in comparison

to the temperature-independent two-layered model (Figure 4.16), the addition

of temperature leads to a remarkable comparison with experimental results,

Figure 4.20. All of the sensitive materials exhibit large values of Ω(2), with the

insensitive materials having very low values. In fact, the separation between

system types is no longer as obvious. NTO is again an exception, and depends

very strongly on the choice of Ω𝑚𝑎𝑥. If the -NO2 rocking modes are placed

within the phonon bath, Ω(2) comes in line with other materials in its sensitivity

class. However, it falls in line with the highly sensitive compounds if Ω𝑚𝑎𝑥 =

200 cm-1 is used. This once again suggests that these modes should be

193

included in the phonon manifold when present at such low frequencies. For

the purpose of the remaining discussion, these rocking modes will therefore

be assumed to form part of the amalgamated phonon bath.

Figure 4.20: Temperature dependent two-layer model to predict impact sensitivity of the

molecular EMs. The equilibrium temperature is set at 300 K and integration of Ω(2) is upper

bound by the limiting frequencies given in Table 4.5. Note NTO(Ω𝑚𝑎𝑥=200) has Ω(2)=120 cm−1.

Lifting these integration restrictions lead to only minor increases in the integrations: HBT (+10);

α-FOX-7 (+3), TATB (+1). The relative ordering therefore remains unchanged.

Overall, it is remarkable that the simplifications made here reproduce the

experimental sensitivity ordering to such an extent. With further investigation

into the intricate interplay of electronic factors, as well as the system

dependent normalisation coefficients, A in Equations 4.3 and 4.4, this model

seems promising for understanding the impact sensitivity of organic EMs. As

no experimental data were available against which to compare variable

temperature predictions, this will not be discussed further here. However, the

model does reflect the qualitative trend of decreasing sensitivity of all

194

compounds as 𝑇 → 0 K, and hence convergence of intrinsic material

sensitivities.

The addition of temperature effects introduces the ability to investigate which

modes can become activated upon initial excitation of the lattice. This is done

by separating the initial temperatures of the phonon bath modes, Tp, to be

different from the remaining modes. In the two-stage construction, this leads

to construction of the doorway mode populations based on initial Tp excitation.

While limited experimental results are available which provide a thorough

study of temperature-dependent sensitivities, it is worth considering a single

example briefly. Due to its popularity 𝛽 -HMX was chosen. In line with

experimental reports for organic molecular crystals, a model phonon ‘shock’

temperature of 2000 K is chosen, Figure 4.21. As can be seen, the relative

rates of up-pumping depend strongly on the frequency in question. Generally,

the doorway modes with Ω𝑚𝑎𝑥 < ω < 2Ω𝑚𝑎𝑥 are (as expected) most highly

excited. Fine-tuning of the model proposed here requires a more fundamental

knowledge of which frequencies are in fact responsible for decomposition

processes.

Figure 4.21: Variable temperature P(Ω(2)) for β-HMX. The initial excitation temperature for the

phonon bath modes (below Ω𝑚𝑎𝑥 = 195 cm-1) is taken to be 2000 K. The temperature of the

remaining states is then plotted over the temperature range 10-300K. Note the phonon bath has

been omitted from the x-axis.

195

Extending this shock model across different systems requires consideration of

the phonon heat capacities, Cph, which can be approximated by assuming

within the Einstein model that each phonon mode contributes 𝑘𝐵 to the heat

capacity at and above ambient conditions. Upon impact with the same energy,

𝑈, the total amount of energy transferred to the material as heat depends on

the compressibility of the material, Equation 3.1. However, without data on the

compressibility of the materials used here, it can be roughly assumed that all

of the molecular materials will behave roughly the same. This is generally a

good approximation, with available ambient pressure bulk moduli of these

materials being very similar (HMX, 14.3 GPa28 ;FOX-7, 12.6±1.4 GPa29 ; TATB,

14.7±0.8 GPa30). It can therefore be assumed that the same proportion of input

energy transforms into heat for these materials. Without knowledge of the

system-dependent Grüneisen parameters, it is not possible to estimate final

bulk equilibrium temperatures. However, for the purpose of the present

discussion, it is sufficient to note that the initial phonon excitation depends on

θ𝑝ℎ = 𝑞/𝐶𝑝ℎ

Equation 4.5

Where 𝜃𝑝ℎ is the phonon quasi-temperature, 𝐶𝑝ℎ is the phonon heat capacity

and q is the heat added to the system. For an arbitrary input energy, the

phonon quasi-temperature therefore decreases with increasing number of

phonon bands.

As arbitrary values, an input energy of 21000 cm-1 is chosen, and corresponds

to the input heat evaluated for a 4 GPa impact on naphthalene (with two

molecules in the primitive cell), and a shock phonon quasi-temperature of ca.

2000 K. 31 This is arbitrarily assigned to be the phonon quasi-temperature of β-

HMX, and the remaining materials scaled accordingly, Figure 4.22. In

construction of the two-layered model in this way, the initial excitation of the

doorway modes occurs via quasi-temperature populations of the phonon bath,

and subsequent up-pumping is also performed using a quasi-temperature

populated phonon bath. The same procedure is done for a β-HMX phonon

quasi-temperature of 1000 K and 3000 K, Figure 4.22. Despite the major

196

approximations, there is again an excellent agreement observed between the

predicted sensitivity ordering, and very similar to that conducted under

equilibrium temperature in Figure 4.20. The same exponential trend is

observed in all cases and suggests consistency within the model. This model

may therefore offer a means to begin to probe the effects of different

experimental conditions across a range of materials. Additional data, including

accurate heat capacities and compressibility (and associated changes in

frequencies), can be added for further refinement of the model.

Figure 4.22: Predicting impact sensitivity using the two-layered model at ambient temperature

of 300 K. The phonon modes are initially excited to shock temperature Tsh. In all cases Ω𝑚𝑎𝑥 for

NTO is 240 cm-1. The y-axes are comparable and reflect an increase in reactivity with increased

shock temperature (and hence stronger impact).

Temperature Effects: Variable Temperature Frequencies

While the introduction of a temperature effect does lead to insight into

interesting phenomena, further parameters are clearly required for its

development. Most crucially is the validity of the underlying vibrational model

that is used in each case. To begin to analyse the underlying vibrational

structure, INS spectra were collected across a range of temperatures as

197

documented in Section 4.4. This offered particular insight into the position of

Ω𝑚𝑎𝑥. Due to time restrictions this was only possible for β-HMX, TATB and α-

FOX-7. However, these represent compounds showing a reasonably broad

range of structural types (e.g. layered, hydrogen bonded, or no directional

intermolecular contacts), and therefore offer a good indication of temperature

effects on vibrational frequencies in molecular materials in general. An overlay

of the INS spectra across the temperatures clearly suggests no notable shift

in the vibrational frequencies of the model organic materials at ω < 800 cm-1

(i.e. within the region of interest for up-pumping calculations), Figures 4.23 and

4.24. Most importantly in terms of the present discussion, the values of Ω𝑚𝑎𝑥,

the relative positions of the major peaks with respect to Ω𝑚𝑎𝑥, and the number

of peaks within the region of interest (< 3Ω𝑚𝑎𝑥) remains unchanged. In β-HMX,

Figure 4.23A, Ω𝑚𝑎𝑥 does not shift between 10-200 K, and the largest shift in

frequency is just ±3 cm-1 on comparing the 200 K and 10 K frequencies. TATB,

Figure 4.23B, however, exhibits a ca. 15 cm-1 decrease in Ω𝑚𝑎𝑥 on heating

from 0 to 200 K. The eigenvector of Ω𝑚𝑎𝑥 reveals this mode to be a rocking

motion of the molecules, perpendicular to the TATB layers. These layers,

stabilised by weak van der Waals interactions, are most susceptible to thermal

expansion. It is therefore logical that this mode should soften on increasing

temperature. The higher frequency modes in TATB do not shift by more than

ca. 4-5 cm-1 on heating from 10 to 200 K. Thus, overall, this imposes two

effects on TATB:

1. The maximum window into which overtone and combination modes can

couple in the first instance decreases by 𝟑 × 𝚫𝛀𝒎𝒂𝒙 (i.e. the change in

𝛀𝒎𝒂𝒙 observed due to thermal heating/cell volume change). This leads

to a change in the upper limit from 𝟑𝛀𝒎𝒂𝒙 ≈ 𝟒𝟔𝟓 → 𝟒𝟐𝟎 cm-1, and

hence exclusion of the vibrational band at ca. 450 cm-1.

2. The rate of up-pumping into these modes will decrease, due to larger

energy separations.

A similar effect can be expected for NTO, which is constructed from a similar

layering motif, Figure 4.1.

198

The INS spectra for α-FOX-7, Figure 4.24, show that between 10 – 150 K,

there is a change of no more than ±2 cm-1 in the internal modes. For this

compound Ω𝑚𝑎𝑥 decreases slightly, from 183 cm-1 at 10 K to 180 cm-1 at 150K.

Figure 4.23: Variable temperature INS spectra for (A) β-HMX and (B) TATB from 10 K to 200 K. All

spectra have been corrected for sample container contributions and background. The intensities

of each system have been normalised to a well-resolved isolated peak (ca. 246 cm-1 for HMX and

ca. 446 cm-1 for TATB). Note that this does introduce errors in the absolute comparison of

intensities across the spectra. The change in intensities across the INS spectra reflects increasing

vibrational amplitude (populations) of these vibrational modes in line with Equation 2.47.

199

Figure 4.24: Variable temperature INS spectra for α-FOX-7 from 10 K to 150 K. All spectra have

been normalised to the peak at ca. 385 cm-1.

It therefore appears that over a broad temperature range, there can be

expected to be only slight changes in the vibrational structure of these

compounds. Hence, the up-pumping model build throughout this chapter can

be expected to largely reflect ambient temperature phenomena. The most

important factor in considering the up-pumping calculations is the placement

of Ω𝑚𝑎𝑥 , which appears to decrease to a notable extent for the layered

compounds. This is expected to affect NTO, and may partially indicate the

inability of this model to accurately reproduce NTO sensitivity. While this may

contribute to the temperature variation in the sensitivity of these compounds,

further work is required to fully understand this phenomenon.

4.5.3.5 Up-Pumping from Zone-Centre Frequencies

It is clear from Section 4.5.2 that negligible band dispersion is observed across

the Brillouin zone of the molecular energetic materials. As many organic

energetic materials are composed of very large molecules in large, low

symmetry unit cells, the approach described above is limited by computational

resources. Much more tractable, however, is the calculation of zone-centre

200

vibrational frequencies. It is therefore interesting to determine whether the

same trends can be obtained from only the zone-centre vibrational structure.

To investigate this, only two of the above models will be discussed, the

overtone pathways as in Figure 4.10, and the two-level combination pathways,

Figure 4.20. The zone-centre phonon DOS do not change drastically with

respect to those of the full phonon DOS in Figure 4.3. If the most promising

overtone-based method is examined (i.e. integration across Ω𝑚𝑎𝑥 < ω <

2Ω𝑚𝑎𝑥), Figure 4.25, it is found that no notable changes occur on moving from

the full phonon dispersion to zone-centre frequencies. The same is true if the

temperature-dependent two-level method is considered, Figure 4.25. The only

notable shift is again NTO, whose Ω(2) appears to increase slightly when only

the zone centre is considered.

Figure 4.25: Vibrational up-pumping based on zone-centre phonon DOS. (Left) calculation of the

temperature independent overtone contribution to N=3. (Right) Two-level system with

equilibrium temperature 300 K. The addition of temperature in the latter is responsible for the

large increase in the y-axis.

201

It is therefore possible to add an additional material to the test set, which was

too large for complete phonon dispersion calculations. Triacetonetriperoxide

(TATP) is a well-known primary explosive, with impact sensitivity < 1 J.25 Its

structure, phonon DOS and Ω(2) are given in Figure 4.26. It is very promising

to find that this material is well placed as a highly sensitive material in both

prediction methods. However, it is best placed as being the most sensitive

compound upon addition of a temperature term, Figure 4.25.

Figure 4.26: (A) Molecular structure of TATP. (B) Crystallographic structure of TATP. (C) Zone-

centre 𝑔(ω) for TATP, and (D) Ω(2), with the restriction that ω2 < Ω𝑚𝑎𝑥.

It follows that, provided the phonon DOS is well reproduced by the zone-center

frequencies – i.e. that the dispersion curves exhibit negligible dispersion – it

may in fact not be necessary to calculate the full vibrational structure. This

202

opens the door to examining very many organic molecular materials, which

may be too large for such expensive calculations. Moreover, it also opens up

the possibility of using standard lab-based Raman or terahertz spectrometers,

to probe the Brillouin zone-centre modes with sufficient detail to offer insights

into impact sensitivity behaviour.

4.6 Conclusions

The organic molecular energetic compounds studied in this chapter span a

broad range of both molecular and crystallographic structure types. From

highly sensitive compounds like TATP, to highly insensitive compounds such

as TATB, the organic EMs exhibit immense diversity in their sensitivity

properties. Analysis of the electronic band structure suggests that no

correlation exists between the size of the electronic band gap and the

sensitivity of these compounds. Hence, the ‘band gap criterion’ fails across the

subset of EMs investigated here.

The full phonon dispersion curves were generated for a series of seven organic

EMs: ABT, HNB, β-HMX, HNT, α-FOX-7, NTO and TATB. Comparison of β-

HMX, α-FOX-7, NTO and TATB to INS spectra suggest that DFT methods

produce excellent agreement with the experimental vibrational structure of

these types of materials.

Based on the vibrational up-pumping model, several qualitative correlations

could be found. Of the rapid, qualitative approaches, the most promising

appears to be simple correlation between the density of doorway modes with

the impact sensitivity. Physically, this can be related to the rate with which the

initial energy can transfer from the excited phonon bath into the internal

vibration manifold. Indeed, if the overtone pathways are considered and

projected onto the doorway region, an excellent correlation is observed with

impact sensitivity.

203

Consideration of the combination pathways by means of the two-phonon

density of states leads to rather poor correlation with experimental sensitivities.

However, this can be largely rectified by implementing a two-layered model.

That is, if explicit consideration is given for the initial excitation of the doorway

modes by the first overtone, and the combination pathways considered

subsequently, then the overall correlation with experiment becomes promising.

However, clear deficiencies remain. As a final step, it is found that the

distribution of doorway modes (i.e clustered at the upper or lower end of the

doorway region) requires some consideration for the relative contribution of

each doorway mode to up-pumping. This was treated by the addition of a

temperature term, populating vibrational states by the Bose-Einstein

populations. If this term is considered, the predicted trend in impact

sensitivities matches very well with experimental values. Hence, the work in

this chapter has developed a fully ab initio approach to ranking the impact

sensitivity of EMs based on a vibrational up-pumping model. In doing so, the

work in this chapter has successfully unified and expanded on competing

models that have been reported in the literature.

Reproducing the models based on zone-centre frequency calculations leads

to the same conclusions. This is presumably due to the low wave vector

dispersion that is exhibited for these materials. Hence, the zone centre

frequencies provide a good representation of the total vibrational structure.

This opens the door to implementing this model to large molecular materials,

such as TATP, for which the task of obtaining complete phonon dispersion

curves is simply too large.

It is remarkable to find that this simple mechanism of up-pumping appears to

provide a means to predict the relative impact sensitivity of a wide range of

structural types. The minor discrepancies that remain are likely due to the

intricacies that surround the differences in phonon scattering rates (i.e.

anharmonic coupling strengths) rates of phonon propagation in these materials,

as well as the dissociation energies of the different molecules.

204

4.7 Suggestions for Further Work

The method presented here has been a first approach at understanding the

up-pumping structure in EMs. Foremost, the work presented here includes

only seven molecular EMs, and therefore a larger number of systems is a clear

direction for additional work. On this subset of organic EMs, the model has

proved promising, despite the numerous simplifications and assumptions that

have been made. Enhancing this model by reducing the significance of these

assumptions is also a clear direction for further work, namely:

• It is clear that these methods perform well at ordering materials with

structurally similar explosophoric moieties. However, better

understanding of the electronic structure and decomposition pathways

is required to compare across different structure types.

• Explicit consideration of anharmonicity constants may prove important

in further resolving differences in predicted sensitivities.

• In this model, the phonon bath has been assumed to remain unchanged

by up-pumping. This is clearly not a realistic assumption. Further work

is required to include this consideration.

• Further investigation is required to unambiguously define the phonon

bath region, particularly in cases such as NTO.

The addition of temperature opens the door to very many possibilities within

the model presented in this chapter. In particular in its ability to introduce a

shock temperature to model the effects of different input energies. However,

an impact is associated with compression of the sample, which has been

neglected in this work. Explicit consideration for the effect of pressure on the

vibrational structure will therefore be a great asset to developing this model

further.

205

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Chapter 5

Vibrational Up-Pumping in Polymorphic Materials

5.1 Introduction

Due to the flexibility of molecules and the low directionality of their weak

intermolecular interactions, molecules can adopt a variety of packing

arrangements upon crystallisation. Hence, the same compound can exist in

different crystal forms, or polymorphs. This phenomenon is widespread across

molecular materials, with Walter McCrone famously stating that “every

compound has different polymorphic forms and the number of forms known for

a given compound is proportional to the time and energy spend in research on

that compound.”1 Different polymorphic forms can be obtained during

crystallization, for example by controlling the speed, temperature or pressure

under which nucleation occurs.2,3 Alternatively, polymorphs can interconvert

upon heating or application of external pressure.4,5 Polymorphic forms are

known to exhibit drastically different physical properties. This phenomenon has

therefore been strongly monitored by the pharmaceutical industry, where

polymorphs can exhibit, for example, different solubilities6 and

compressibilities.7 Hence, a drug composition that is prepared based on one

polymorph can behave differently than that based on another polymorphic form.

Energetic materials (EMs) are also highly prone to polymorphism. Some of the

most well-known EMs exhibit rich polymorphism. For example, RDX is known

to exist in at least two polymorphic forms under ambient conditions, the α- and

β-forms, and two additional forms can be obtained under hydrostatic pressures,

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γ-RDX at ca. 5.2 GPa8 and δ-RDX at ca. 17.8 GPa. A fifth form, ϵ-RDX, has

also been identified on simultaneous application of heat and pressure.9 Other

well-known EMs, such as CL-20, TNT, ammonium perchlorate and ammonium

nitrate, also exhibit rich polymorphism. As polymorphic transformations often

occur when a material is exposed to extremes of temperature and pressure,

their occurrence is crucial to understanding the detonation pathways of EMs.

Despite knowledge of their existence, most EM literature does not explicitly

consider polymorphic modifications during impact testing. A notable exception

is HMX, Figure 5.1. Under ambient conditions HMX exists in the monoclinic β-

form. The orthorhombic α-form can be obtained from recrystallisation under

elevated temperatures and is stable between 377 and 429 K, and can be

recovered to ambient conditions.10 The δ-form is obtained by heating the β-

form, and is stable above 429 K.11 While highly metastable, the δ-form can be

recovered to ambient conditions upon quench cooling.10 A fourth form, a

hydrate (often denoted γ-HMX) is also known to be readily prepared under

ambient conditions on rapid recrystallisation of β -HMX from aqueous

solutions.10

The absolute sensitivity of the δ-form is open to debate, although it is accepted

to be considerably more sensitive to impact than the β-form.12 A thorough

analysis of reported sensitivities by Cady and Smith,10 and subsequent work

by Scott,13 suggests δ -HMX to have a comparable impact sensitivity to

pentaerythritol tetranitrate (PETN), the highest sensitivity secondary explosive

(ca. 3 J)14 in common use. Other reports have suggested δ-HMX to be as

sensitive as lead azide (< 1 J)15 or other primary explosives.16 While the exact

level to which δ-HMX is more sensitive than the β-form remains uncertain, it is

clear that they do exhibit very different sensitivity properties. The available

literature therefore suggests the impact sensitivity ordering for HMX as δ >

γ > α > β.10

To date, no high-pressure forms of HMX have been structurally characterised.

However, some experiments do suggest two potential high-pressure

transitions under quasi-hydrostatic conditions.17,18 A conformational transition

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(i.e. associated with a small change in the molecular geometry, without a large

change in the crystallographic structure) was observed at ca. 12 GPa, with an

abrupt phase transition observed above 27 GPa. 18

Figure 5.1: Structure of HMX. (A) Schematic representation of the HMX molecule. (B) Crystal

structure of 𝛽-HMX 𝑃21/𝑐 (CCDC Ref: OCHTET01). (C) Crystal structure of 𝛿-HMX 𝑃61 (CCDC Ref:

OCHTET03). Atoms are coloured as (red) oxygen, (blue) nitrogen, (grey) carbon, and (white)

hydrogen.

FOX-7 is also well known to be polymorphic. Under ambient conditions, FOX-

7 is stable as the monoclinic 𝛼-phase. Upon heating, two phase transitions

occur, first to β-FOX-7 at 111 °C, and subsequently to γ-FOX-7 at 173 °C.19

These transitions correspond to an increased layering of the FOX-7 molecules,

Figure 5.2. Spectroscopic investigations and thermal analysis have suggested

a fourth high-temperature phase (δ-FOX-7) to exist immediately prior to the

decomposition temperatures (i.e. at T > 230 oC), although no crystal structure

213

has yet been obtained. Some reports suggest that the δ-phase is simply a

decomposition product of the γ-phase.20 The application of pressure has also

been demonstrated to yield a variety of polymorphic transformations, including

formation of two new phases on compression of α-FOX-7 to ca. 2 GPa and 5

GPa,19,21 denoted the α ’ and ϵ phases, respectively. The latter has been

suggested to play a role in the shock-induced processes of FOX-7.20

It is generally accepted that energetic materials that adopt layered crystal

packing (e.g. TATB) display lower sensitivities, although no underlying

mechanism for this effect has been deduced. Two common theories include:

1 Increased hydrogen bonding, which forms upon layering, leads to

dissipation of energy through the layers22

2 Slip planes associated with layered materials23,24

Some have also proposed that a higher inter-layer compressibility of layered

materials effectively decreases the energy input from an impact. However, in

accordance with Equation 3.1, a larger compressibility in fact does quite the

opposite, and leads to a larger proportion of the impact energy converting into

heat, rather than work. This has the effect of enhancing the direct excitation of

the molecules in the system. In line with the theory proposed in Chapter 4,

higher compressibility also leads to hardening of vibrational modes and thus

enhances up-pumping. Instead, the lower compressibility of hydrogen-bonded

layers has instead been proposed to account for the lower sensitivity of layered

materials. 22

The polymorphs of FOX-7 therefore offer an opportunity to test the up-pumping

model developed in this thesis, on the effects of how layered structures

influence impact sensitivity.

214

Figure 5.2: Structure of FOX-7. (A) Schematic representation of the FOX-7 molecular structure. (B)

Crystal structure of α -FOX-7 𝑃21/𝑛 (CCDC Ref: SEDTUQ03). (C) Crystal structure of β -FOX-7

𝑃212121(CCDC Ref: SEDTUQ06). (C) Crystal structure of γ-FOX-7 𝑃21/𝑛 (From Crawford et al 25).

The angle between FOX-7 molecules (∠(FOX-7)) is given in each case. Atoms are coloured as (red)

oxygen, (blue) nitrogen, (grey) carbon and (white) hydrogen.

5.2 Aims

The work in this chapter seeks to use the model employed in Chapters 3 and

4 to investigate two polymorphic systems: HMX and FOX-7. The former

contains two polymorphs that are well-known to exhibit very different impact

sensitivities, while the impact sensitivities of the latter have yet to be

experimentally reported. This chapter therefore aims to:

• Determine whether the up-pumping model is sensitive to polymorphic

modifications

215

• Experimentally determine the impact sensitivity of the 𝛾-polymorph of

FOX-7

• Explore the effects of increased layering on the sensitivity of energetic

materials

• Rationalise the sensitivity of FOX-7 polymorphs based on the up-

pumping model.

5.3 Materials

Synthesis of FOX-7. Preparation of 1,1-diamino-2,2-dinitroethene (DADNE,

FOX-7) followed from Scheme 5.1.26 2-methyl-4,6-pyrimidinedione (6.0 g, 0.05

mol) was dissolved in concentrated H2SO4 (95%, 45 mL), at temperatures <

30 oC. Concentrated HNO3 (99%, 20 mL) was subsequently added dropwise

maintaining temperatures < 20 oC, and the sample stirred for 3 hours. The

resulting material was rinsed in concentrated H2SO4, added to deionized water

(100 mL) and stirred for 2 hours. The resulting product was analysed by 1H

and 13C NMR, and subsequently by X-ray powder diffraction.

Scheme 5.1: Synthetic approach for preparation of FOX-7.

Preparation of 𝛾-FOX-7. The metastable 𝛾-form was prepared by heating the

𝛼-form to 180 oC for approximately 2 hours, and quench cooling the material

to ambient conditions. The phase purity was confirmed by X-ray powder

diffraction.

216

Preparation of 𝛿-HMX. A sample of β-HMX was slowly heated in a furnace to

190 oC,27 and held at this temperature for ca. 12 hours. The sample was

quench cooled with an ice bath, and the polymorphic phase verified by X-ray

powder diffraction. The sample was stored under an N2 atmosphere for

approximately 1 day prior to INS analysis.

BAM Drop Hammer Testing. BAM fall hammer testing (BFH-12) was

conducted at the Cavendish Laboratory, University of Cambridge. A sample of

ca. 40 mm3 was placed in an anvil device and sealed between two co-axial

steel cylinders. The anvil components were disposed of between each sample

test. In this work, the Limiting Energy (1-in-6) go/no-go criterion was used;28 a

‘go’ was considered when a flash, audible explosion or discolouration of the

sample (black) or scorch marks on the anvil was observed.

X-ray Powder Diffraction. All solid samples were analysed by powder X-ray

diffraction using a D2 PHASER diffractometer, with Cu 𝐾𝛼 radiation (𝜆 =

1.5406 Å). Data were collected in Bragg-Brentano geometry over the range 2𝜃

= 5-500 (d-spacing ≈ 1.8-17 Å).

Inelastic Neutron Scattering Spectroscopy. Inelastic neutron scattering spectra

were collected on the TOSCA29,30 spectrometer at the ISIS Neutron and Muon

Facility. Samples (ca. 1.5 g) were placed in aluminium sample holders. The

samples were cooled to ca. 10 K and spectra collected for a total of ca. 400

μAh. Data from the forward- and backward-scattering detectors were summed

and corrected for scattering from the sample container and background. Data

processing was performed using the Mantid software.31 Simulated INS spectra

were generated using ABINS32 as implemented in Mantid.

Condensed Matter Calculations. All condensed matter vibrational calculations

were based on experimental crystal structures. Structures for δ -HMX

(OCHTET03) and β-FOX-7 (SEDTUQ06) were sourced from the Cambridge

Crystallographic Data Centre (CCDC). The input structure for γ-FOX-7 was

taken from Crawford et al.25 Data for β-HMX and α-FOX-7 were taken from

Chapter 4. All structures were optimised using plane-wave DFT (pw-DFT) as

217

implemented in CASTEP v16.33 The GGA functional of Perdew-Burke-

Ernzerhof34 (PBE) was used along with Grimme’s D2 dispersion correction

scheme,35 PBE-D2. This scheme has previously been demonstrated to

perform well for these materials.21,36 The electronic wavefunction was

expanded in plane waves to a kinetic energy cut-off of 1800 eV (α-FOX-7),

1300 eV (β-HMX, δ-HMX and β-FOX-7) and 950 eV (γ-FOX-7). All forces were

converged to < 5 × 10−4 eV/Å and stresses to < 5 × 10−4 GPa. γ-FOX-7 was

converged to less stringent parameters, forces < 1 × 10−3 eV/ Å and stresses

to < 1 × 10−3 GPA. The energy change per atom was accepted after

convergence < 5 × 10−9 eV/atom for all cases except γ-FOX-7, for which a

convergence was accepted < 1 × 10−8 eV/atom. The electronic structure was

sampled on a k-point Monkhorst-Pack37 (MP) grid with spacing no more than

0.05 Å-1. Norm-conserving pseudopotentials were used throughout. Optimised

geometries are summarized in Table 5.1.

Table 5.1: Optimisation Parameters for the HMX and FOX-7 materials investigated here.

a b c 𝛂 𝛃 𝛄 V 𝚫𝐕

β-HMXexp 6.525 11.024 7.362 90 102.642 90 516.675 +4.00%

β-HMXcalc 6.624 11.256 7.373 90 102.222 90 537.299

δ-HMXexp 7.711 7.711 32.553 90 90 120 1676.270 +0.57%

δ-HMXcalc 7.670 7.670 33.091 90 90 120 1685.860

α-FOX-7exp 6.934 6.622 11.312 90 90.065 90 519.470 +2.20%

α-FOX-7calc 7.089 6.623 11.440 90 91.273 90 530.898

β-FOX-7exp 6.974 6.635 11.648 90 90 90 538.943 +1.12%

β-FOX-7calc 7.093 6.495 11.830 90 90 90 544.967

γ-FOX-7exp 13.354 6.895 12.050 90 111.102 90 1035.110 +5.02%

γ-FOX-7calc 13.565 7.084 12.224 90 112.247 90 1087.170

Phonon calculations were performed using the linear response method as

implemented in CASTEP v16 at either the -point or sampled across the

Brillouin zone on a regular grid of wave vectors and subsequently Fourier

interpolated onto a finer grid. Electronic band structures were calculated using

localized basis sets (H- H_pob_TZVP_201238; C- C_m-6-

311G(d)_Heyd_200539; N- N_m-6-311G(d)_Heyd_200539; O- O_m-6-

311G(2d)_Heyd_200539) within the CRYSTAL17 code,40 based on

218

experimental structures. Band structures were generated using the HSE06,39

B3PW9141 and PBE34 functionals. Electronic structure was calculated across

no less than 120 k-points, evenly spaced across an MP grid. The wavefunction

was accepted after the absolute change in SCF cycle energies was < 10-8. For

all materials, the tolerances (TOLINTEG) were set at 7 7 7 9 30 (as

recommended for use with these basis sets39).

5.4 Results and Discussion

5.4.1 Polymorphism of HMX

5.4.1.1 Electronic Structure of HMX Polymorphs

The calculated band gaps for both β - and δ -HMX were found to be

approximately the same, with the δ-form exhibiting a slightly larger band gap

than the β-form, Table 5.2. Furthermore, δ-HMX exhibits an indirect band gap

for all three functionals, while two of the three functionals suggest that β-HMX

has a direct band gap. Any correlation to the ‘band gap criterion’42 (i.e. that the

more sensitive material has the smaller band gap) therefore does not hold

when considering these two polymorphs of HMX.

With no notable electronic differences in the solid state, it is therefore worth

considering a vibrational basis to rationalise the sensitivity differences.

Table 5.2: Band gaps for δ-HMX as compared to those for β-HMX. Band gaps are indicated as

direct (D) or indirect (I).

B3PW91 PBE HSE06

β-HMX 5.4954 (D) 3.6826 (I) 5.2176 (D)

δ-HMX 5.7422 (I) 3.7011 (I) 5.4745 (I)

219

5.4.1.2 Vibrational Structure of HMX Polymorphs

The primitive unit cell of 𝛿-HMX is considerably larger than that of 𝛽-HMX, and

it was therefore not possible to obtain a full phonon dispersion curve for the

former. However, as demonstrated in Section 4.5.3.5, for the materials which

exhibit negligible dispersion, the zone-centre frequencies are sufficient for

consideration of the vibrational up-pumping model. Given the low dispersion

of the 𝛽-polymorph (see Chapter 4.5.2), it is reasonable to expect a similar

character in the 𝛿-form.

Figure 5.3: Inelastic neutron scattering spectra of δ-HMX at ca. 10 K. (Top) The experimental

pattern and (Bottom) simulated patterns are given. The latter is generated from Γ -point

frequencies only. The vertical dotted line indicates Ω𝑚𝑎𝑥 in each case.

The zone-centre vibrational structure for 𝛿-HMX was therefore calculated, and

compared to experimental INS data, Figure 5.3. There is generally very good

agreement between the simulated and experimental frequencies. The value of

Ω𝑚𝑎𝑥 sits only ca. 5 cm-1 lower in the experimental pattern as compared to the

simulated pattern, with the close comparison suggesting little band dispersion

exists at the top of the phonon bath. The frequencies are very well reproduced

220

across the INS spectrum, underestimated by only 2-3% across most modes,

Table 5.3. A notable exception is 𝑀5, which is underestimated by ca. 6% in the

simulated pattern. This corresponds to the deformation modes of the HMX ring,

and suggests that PBE-D2 may struggle in reproducing some internal modes

of these materials, which is also apparent when the higher frequency modes

with ω > 1000 cm-1 are considered. This was also noted in Chapters 3 and 4

for the internal frequencies of other materials. Overall, however, it appears that

the model used is in general a good reproduction of the experimental

vibrational structure for δ-HMX.

Table 5.3: Comparison of simulated (-point only) and experimental INS frequencies for well-

resolved peaks. The difference is given as a percentage over the experimental value.

INS Calc. Δω /%

Ω𝑚𝑎𝑥 178 177 -0.56%

𝑀1 201 198 -1.49%

𝑀2 222 218 -1.80%

𝑀3 238 230 -3.48%

𝑀4 253 250 -1.19%

𝑀5 331 312 -5.74%

𝑀6 374 367 -1.87%

𝑀7 390 382 -2.05%

𝑀8 404 388 -3.96%

𝑀9 418 405 -3.01%

𝑀10 448 436 -2.68%

𝑀11 471 461 -2.12%

𝑀12 587 561 -4.43%

𝑀13 649 631 -2.77%

𝑀14 710 688 -3.10%

𝑀15 735 703 -4.35%

𝑀16 753 730 -3.05%

𝑀17 837 820 -2.03%

𝑀18 924 908 -1.73%

𝑀19 1026 970 -5.46%

221

The phonon density of states, 𝑔(ω), and two-phonon density of states, Ω(2) =

𝛿(𝜔 − 𝜔1 − 𝜔2), are given in Figure 5.4, under the restriction 𝜔2 < Ω𝑚𝑎𝑥. The

value of Ω𝑚𝑎𝑥 is placed at 160 cm-1 in δ-HMX, which agrees well with both the

theory and INS spectra. Based on analysis of the 𝑔(ω), an alternative would

be to place it at 260 cm-1, above the ring deformation modes that span the

region 160 < ω < 260 cm-1. However, no evidence exists to suggest this is a

more appropriate placement of Ω𝑚𝑎𝑥, and the former will subsequently be used.

While Ω𝑚𝑎𝑥 is found to be lower for δ-HMX (160 cm-1) than in the β-form (195

cm-1), the doorway region (Ω𝑚𝑎𝑥 < ω < 2Ω𝑚𝑎𝑥) is notably denser in the former

(7.1 vs 4.4 states per atom). Hence it can already be suggested that the δ-form

will be more readily excited by vibrational up-pumping. Furthermore, the onset

of Ω(2) occurs approximately 50 cm-1 earlier in δ-HMX, and grows much more

rapidly than for β-HMX. Qualitatively, all of these factors suggest the δ-form to

be more sensitive according to the up-pumping model.

To compare the two polymorphs, vibrational up-pumping was considered in

line with the two most promising models from Chapter 4, but now applied to -

point data only: (1) the contribution of the first two overtones to the region

Ω𝑚𝑎𝑥 → 3Ω𝑚𝑎𝑥, and (2) the two-level model under an equilibrium temperature

of 300 K. Note that as in Chapter 4, the lack of a specific target frequency

requires consideration of an ‘indirect’ phonon up-conversion mechanism.43

Hence up-pumped values are normalised by ∫𝑔(ω).

In the first model, the two lowest order overtones (i.e. the fastest coupling

pathways) are generated, Figure 5.5, and their projection onto 𝑔(ω) are

considered, i.e. 𝑃(𝑔(ω)), which is then normalized by ∫𝑔(ω). As only the first

two overtones are included, this restricts the upper value for integration to

3Ω𝑚𝑎𝑥, with the resulting ∫𝑃(𝑔(ω)) for δ-HMX (~6.62 a.u.) > β-HMX (~5.85

a.u.). Thus the overtone up-pumping model suggests more pathways exist for

the δ-form. As such δ-HMX is therefore predicted to be more sensitive to

impact than the β-form, and ranks the sensitivities as HNB > δ-HMX ≈ TATP >

ABT > β-HMX according to Figure 4.25.

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Figure 5.4: Comparison of the vibrational structure of two HMX polymorphs. (Top) 𝑔(ω) and Ω(2)

for β-HMX, and (bottom) 𝑔(ω) and Ω(2) for δ-HMX. 0→ Ω𝑚𝑎𝑥 and Ω𝑚𝑎𝑥→ 2Ω𝑚𝑎𝑥 are identified

in 𝑔(ω) by yellow and purple, respectively. Note that Ω(2) have been normalised by ∫𝑔(ω) to

account for a different number of atoms (and hence vibrational modes) in the unit cell according

to the indirect up-pumping mechanism.

223

Figure 5.5: Overtone contributions to vibrational up-pumping in HMX polymorphs. The overtones

are shown for (blue) 𝑁 = 2 and (green) 𝑁 = 3, and overlain by 𝑔(ω) (black).

Within the two-layer model, the populations from the first overtone (with T =

300 K) are projected onto the doorway region, and up-pumped with the

underlying phonon modes via combination pathways. As Ω(2) (see Figure 5.4)

does not drop to zero < 1800 cm-1 for either polymorph, up-pumping

contributions to ω < 1800 cm-1 are considered for both. This results in ∫Ω(2)

for δ -HMX (~200 a.u.) > β -HMX (~32 a.u.), which again reproduces the

experimental observation that δ-HMX is more sensitive than the β-form. For

the two-layer model the sensitivity ordering is TATP ≈ δ-HMX > HNB > ABT >

β -HMX according to Figure 4.25, which does appear to place δ -HMX

somewhat higher in sensitivity than expected experimentally. However, it

clearly places it in line with the other highly impact-sensitive compounds.

To summarise, it is clear that both up-pumping models predict that the δ-form

should be notably more sensitive to impact than the β-form, consistent with

experimental reports. Thus, while this is a limited test set, it does suggest that

the vibrational up-pumping model is sensitive not only to different molecular

materials, but also to different crystallographic forms of the same energetic

molecule.

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5.4.2 Polymorphism of FOX-7

Having demonstrated the ability of the up-pumping model to distinguish

between different polymorphic forms of HMX, the three temperature-related

FOX-7 polymorphs were considered next.

5.4.2.1 Experimental Impact Sensitivity

Samples of both α- and γ-FOX-7 were subjected to impact sensitivity testing

using the BAM fall-hammer with ‘go/no-go’ criteria. A sample of α-FOX-7 was

first subjected to testing, Table 5.4, with an impact sensitivity of ca. 8 J

according to the 1/6 Limiting Impact sensitivity criteria. This is clearly more

sensitive than is reported based on the ℎ50 statistics (𝐸50 ≈ 24-30 J)44. The

1/6 method indicates the lower end of the sensitivity sigmoidal curve shown in

Chapter 2, and hence is always lower than the ℎ50 value. Moreover, it is known

that FOX-7 sensitivity increases with decreasing particle size, with a limiting

value of ca. 10-11 J having been reported in literature.44

Table 5.4: BAM impact sensitivity results for α-FOX-7. A ‘go’ is indicated as ✓ and ‘no-go’ by .

Trial Number

Mass /kg Height /cm Energy /J 1 2 3 4 5 6

5 22.4 11.20 ✓

17.8 8.90 ✓

1 89.1 8.91 ✓

79.4 7.94 ✓

70.8 7.08

A sample of α-FOX-7 was subsequently heated to convert it to γ-FOX-7. The

α → γ phase transition is a single-crystal to single-crystal transformation.25

Hence, the particle size distribution can be taken to be the same across the

two polymorphic forms, on the basis that no reconstructive phase transition

occurs. When this sample was subjected to BAM hammer testing, the impact

sensitivity was found to be the same to within the experimental error of the

measurement, Table 5.5. This was unexpected given the extreme insensitivity

225

of other layered compounds,45 including those based on the same energetic

material.46

Table 5.5: BAM impact sensitivity results for γ-FOX-7. A ‘go’ is indicated as ✓ and ‘no-go’ by .

Trial Number

Mass /kg Height /cm Energy /J 1 2 3 4 5 6

5 31.6 15.80 ✓

28.2 14.10 ✓

25.1 12.55 ✓

22.4 11.20 ✓

20 10.00 ✓

1 89.1 8.91 ✓

79.4 7.94

Powder samples that did not initiate on impact were therefore analysed by X-

ray powder diffraction (XRPD), Figure 5.6. It was surprising to find that all of

the material that had been subjected to BAM hammer testing had converted to

the α-form, while the material that had not, remained as the γ-form for a period

of at least three days (under ambient conditions) following the experiments,

Figure 5.6. This suggests that upon impact, there is a γ → α phase transition,

which is presumably responsible for the similarities between the impact

sensitivity of the layered and non-layered polymorphs. Further work is required

to determine the cause of this transition: pressure, shear, temperature or their

combination. This demonstrates a clear deficiency in the applications of BAM

hammer testing to the study of polymorphic materials. Upon impact, phase

transitions may be possible. Thus, the material which is actually analysed

during an experiment may not be the intended material and can therefore lead

to potentially erroneous reports and conclusions. Without a clear method to

determine the sensitivity of the γ-polymorph experimentally, the three phases

of FOX-7 were therefore analysed in the framework of the vibrational up-

pumping models developed in this thesis.

226

Figure 5.6: XRPD profiles for FOX-7 before and after BAM hammer treatment. Experimental

patterns (black) are compared to simulated (blue) 𝛼-FOX-7 and (green) 𝛾-FOX-7 in all cases. Note

a small offset in the position of the experimental peak of 𝛾 -FOX-7 at 𝑑 ≈ 5.5 Å, which

corresponds to the crystallographic (0 0 2) plane. This offset is due to minor misalignment of the

diffractometer and sample geometry.

227

5.4.2.2 Electronic Structure

Despite the failings of the ‘band gap’ criterion across the series of molecular

energetic compounds in Section 4.5.1, it was nevertheless worthwhile

considering this effect for the polymorphic forms of FOX-7. The electronic

structure of the three polymorphs of FOX-7 were therefore calculated using

the same three functionals as in Section 4.5.1 and 5.4.1.1. and are

summarised in Table 5.6. There does appear to be a very slight increase in

the band gap as the material becomes increasingly layered.

Table 5.6: Band gap values for the three polymorphs of FOX-7. Band gaps are labelled as direct

(D) or indirect (I) in each case.

Material B3PW91 PBE HSE06

𝛼-FOX-7 3.9833 (I) 2.4483 (I) 3.6719 (I)

𝛽-FOX-7 4.2252 (I) 2.6643 (I) 3.9169 (I)

𝛾-FOX-7 4.2462 (I) 2.5866 (D) 3.9430 (I)

5.4.2.3 Vibrational Up-Pumping in FOX-7 Polymorphs

Note that the phonon dispersion curve for 𝛾-FOX-7 was calculated by Ms S Piggott (Master’s student,

EaStCHEM School of Chemistry, University of Edinburgh).

It is hence worth considering the sensitivity of the polymorphs within the up-

pumping models of impact sensitivity that have been built in Chapters 3 and 4.

The phonon dispersion curves and associated density of states for all three

polymorphs of FOX-7 are given in Figure 5.7. Note the imaginary frequency

associated with the two lowest acoustic branches at q-point B (-0.5 0 0). This

vector runs perpendicular to the FOX-7 planes. As γ -FOX-7 is highly

metastable, further work is required to determine the validity of this result,

although it is not expected to have any notable consequence on the following

discussion. As the FOX-7 layers become increasingly planar, the energy gap

between Ω𝑚𝑎𝑥 and the doorway mode region increases, with Ω𝑚𝑎𝑥 found at

185 cm-1, 170 cm-1 and 160 cm-1 for the α-, β- and γ-polymorphs, respectively.

228

This decrease in Ω𝑚𝑎𝑥 is due to softening of the NO2 rocking modes that are

polarized perpendicular to the FOX-7 planes and therefore soften upon

layering. Within the framework of vibrational up-pumping, this has the effect of

reducing the density of the doorway manifold, Ω𝑚𝑎𝑥 < ω < 2Ω𝑚𝑎𝑥 (see Figure

5.8) and hence greatly reducing the number of potential up-pumping pathways.

Indeed, the doorway density decreases in the sequence α -FOX-7 (~4.67

states per atom) > β-FOX-7 (~4.20 states per atom) > γ-FOX-7 (~2.52 states

per atom).

Figure 5.7: Phonon dispersion curves for three polymorphs of FOX-7. The gap between the

phonon bath and doorway modes is highlighted with a green box.

229

Figure 5.8: Comparison of (left) 𝑔(ω) and (right) Ω(2) for the three polymorphs of FOX-7. The

phonon bath and doorway mode regions in 𝑔(ω) are indicated in yellow and purple, respectively.

Note that Ω(2) have been normalized by ∫𝑔(ω) to account for a different number of atoms (and

hence vibrational modes) in the unit cell according to the indirect up-pumping mechanism.

230

Figure 5.9: INS spectrum (10 K) for γ-FOX-7. The (black) experimental spectrum is given alongside

the (blue) simulated spectra using increasing densities of wave vectors for the phonon

calculations. Note that only first order quantum events are simulated.

231

While it was not possible to isolate the β-form, the γ-form could be quench

cooled, and analysed by INS spectroscopy, Figure 5.9. The calculated

vibrational structure for γ -FOX-7 yields a simulated INS spectrum that

generally agrees well with the experimental spectrum. The most notable

difference is the underestimation of Ω𝑚𝑎𝑥 in the simulated spectrum (Ω𝑚𝑎𝑥 is

160 cm-1 from simulation, and 171 cm-1 from INS). While this is likely to have

some consequence on the calculation of the up-pumping model, it is important

to note that the experimental Ω𝑚𝑎𝑥 is still approximately 20 cm-1 lower than the

experimental Ω𝑚𝑎𝑥 value for α-FOX-7 presented in Chapter 4. The remainder

of the INS spectra exhibit the same expected features, with discrepancies in

the calculated frequencies < 6%. While this does suggest some difficulty with

reproducing the vibrational structure of γ-FOX-7, it is overall representative of

the experimental frequencies and is therefore carried forward for data

processing in the up-pumping model.

As described for the HMX polymorphs, the FOX-7 polymorphs were analysed

within the framework of the two most successful models of Chapter 4. This first

required generation of 𝑔(ω) (for this system, sampling could be obtained from

across the Brillouin zone) and Ω(2) for the three polymorphs, Figures 5.8,

respectively. Across all three polymorphs, Ω(2)adopts a very similar structure.

The onset wavenumber is approximately 270 cm-1 in each case, which reflects

the generally similar structure of the doorway modes. Furthermore, all three

polymorphs exhibit Ω(2) = 0 just above 1000 cm-1, and feature the same

groupings of density. With very similar vibrational structures, it follows that the

main difference in understanding the up-pumping between these polymorphs

will be Ω𝑚𝑎𝑥 and the relative rates at which energy can up-pump into these

nearly identical structures.

If the three FOX-7 polymorphs are analysed first using the overtone-based

model, built on the first two overtones, Figure 5.10, the relative sensitivity

ordering that was determined by doorway density is recovered: α-FOX-7 ≈ β-

FOX-7 > γ-FOX-7, with the overtone up-pumped density equal to ~ 6.0, ~ 6.1

and ~ 4.9, respectively. This places the two FOX-7 polymorphs that exhibit

232

herringbone packing at approximately the same sensitivity, with the layered γ-

form predicted to have a lower sensitivity.

Figure 5.10: Overtone contributions to vibrational up-pumping in FOX-7 polymorphs. The

overtones are shown for (blue) 𝑁 = 2 and (green) 𝑁 = 3, and overlain by 𝑔(ω) (black). Plots are

shown for (A) α-FOX-7, (B) β-FOX-7 and (c) γ-FOX-7.

It is finally worth considering the three FOX-7 polymorphs using the two-

layered model, and an equilibrium temperature of 300 K. Despite the projection

of the overtone pathways onto the doorway modes leading to considerably

fewer doorway contributions for γ-FOX-7, Figure 5.8, the larger number of

populated modes available in the phonon bath for this polymorph greatly

reduces this effect. According to discussions in Chapter 4, all three polymorphs

exhibit Ω(2) = 0 at approximately 1000 cm-1 (Figure 5.8). Hence, this is taken

233

to be the upper limit for integration in the two-layered model. The values of

∫Ω(2) for the three polymorphs are found to be 14.5, 12.6 and 12.2 a.u. for the

α-, β- and γ-polymorphs, respectively. The results of Chapter 4 (Figure 4.20)

suggest that this corresponds to a sizeable difference in predicted sensitivity

between the α- and γ-forms, with the latter being less sensitive.

Both the overtone and the two-layered models therefore suggest that flattening

the crystal layers of an energetic compound should decrease the impact

sensitivity. The models point towards the sensitivity decrease being purely the

result of a decrease in Ω𝑚𝑎𝑥 , given that the remaining 𝑔(ω) are largely

unchanged, Figure 5.8. Hence, a new mechanism for the decreased sensitivity

of layered materials is therefore proposed based on the decreased ability of

these materials to up-pump vibrational energy. Without a larger dataset it is

not yet possible to correlate the change in predicted impact sensitivity to an

absolute change in initiation energy. However, based on the datasets in

Chapter 4, it is predicted that γ-FOX-7 should be notably less sensitive than

the α -form. Thus this model suggests a major deficiency in the current

experimental approach to studying impact sensitivity of polymorphic materials.

Furthermore, this demonstrates the importance of considering mechanically-

induced structural transformations that may occur immediately before, or

during, initiation events.

5.5 Conclusions

Polymorphism is very prevalent amongst energetic materials, and can lead to

drastic changes in a material’s sensitivity to impact. Most notable are the 𝛿-

and 𝛽-polymorphs of HMX. The former has been reported to be as sensitive

to impact as a primary explosive material, while the latter exhibits much lower

sensitivity to impact. Application of the up-pumping model was able to

reproduce these experimental findings and assessed δ-HMX as being a highly

sensitive material. This therefore demonstrates that the up-pumping model is

234

sensitive to polymorphic modifications. It was therefore applied to a second

polymorphic energetic material, FOX-7.

Under ambient conditions, FOX-7 exists in the α -form, which adopts a

herringbone-type structure. When heated, these layers flatten, and are nearly

flat in the γ-form. This form was recovered to ambient conditions and its impact

sensitivity measured using a BAM fall hammer. This suggested that the

layered γ-form had the same impact sensitivity as the α-form, despite the

general principle that layered materials are insensitive. X-ray powder

diffraction, however, revealed that the γ-form undergoes transformation to the

α-form on impact, and hence it is not known which polymorphic phase was in

fact tested. The mechanism for this impact-induced transformation is not yet

known, and may be the result of pressure, temperature or their combination.

The up-pumping model was therefore applied to the FOX-7 polymorphs. Both

the overtone-based model and the temperature-dependent two-layered model

suggested that the layered γ-form should be notably less sensitive than the α-

form. It is suggested that the reduction in sensitivity is the result of a decrease

in Ω𝑚𝑎𝑥 (an observation noted in both the INS spectra and simulated phonon

density of states plots) that results from the increased layering. This reduction

in Ω𝑚𝑎𝑥 is observed across all layered materials studied thus far. Hence, a new

structurally-based mechanism for the decreased sensitivity of layered

materials has been proposed.

Due to the γ → α transition, BAM hammer testing appears incapable of directly

measuring the impact sensitivity of the γ -form. Current experimental

approaches are inadequate for the investigation of polymorphic materials.

Furthermore, this transition demonstrates the importance of considering

structural transformations during the initiation process of EMs, and the ability

of the up-pumping model to assist in the interpretation of experimental results.

235

5.6 Suggestions for Further Work

It is clear from this chapter that the up-pumping models can be applied to

polymorphic series. However, the sample size used here is limited. It is

therefore of great interest to extend this work to a broader set of polymorphic

materials. To do this, it will be necessary to conduct experimental

investigations on the sensitivity of polymorphic materials, many of which have

yet to be thoroughly analysed.

A more detailed analysis of the impact-induced polymorphism of γ-FOX-7 can

also be suggested, with the aim of identifying the mechanism for the γ → α

transition. This can be done by monitoring the material during impact by

spectroscopic or X-ray techniques. Understanding this transformation may be

critical to fully rationalise the transformation that was observed in this work.

Additionally, it will be worth considering the role of the ϵ-form (which forms

when FOX-7 is exposed to pressure21) in this transformation, and its impact

sensitivity relative to the α- and γ-forms.

Furthermore, it is apparent that a deeper correlation between the predicted

impact sensitivity and experimental values must be sought. This can only be

obtained by expanding the set of compounds studied using these models.

However, this also requires accurate capture of the experimental impact

sensitivities of EMs, which can prove difficult in many cases.

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(39) Heyd, J.; Peralta, J. E.; Scuseria, G. E.; Martin, R. L. Energy Band Gaps and Lattice Parameters Evaluated with the Heyd-Scuseria-Ernzerhof Screened Hybrid Functional. J. Chem. Phys. 2005, 123 (17), 174101.

(40) Dovesi, R.; Orlando, R.; Erba, A.; Zicovich-Wilson, C. M.; Civalleri, B.; Casassa, S.; Maschio, L.; Ferrabone, M.; De La Pierre, M.; D’Arco, P.; et al. CRYSTAL14: A Program for the Ab Initio Investigation of Crystalline Solids. Int. J. Quantum Chem. 2014, 114 (19), 1287–1317.

(41) Tran, F.; Blaha, P.; Schwarz, K.; Novák, P. Hybrid Exchange-Correlation Energy Functionals for Strongly Correlated Electrons: Applications to Transition-Metal Monoxides. Phys. Rev. B - Condens. Matter Mater. Phys. 2006, 74 (15), 155108.

(42) Zhu, W.; Xiao, H. First-Principles Band Gap Criterion for Impact Sensitivity of Energetic Crystals: A Review. Struct. Chem. 2010, 21 (3), 657–665.

(43) Dlott, D. D. Multi-Phonon up-Pumpng in Energetic Materials. In Overview of Recent Research on Energetic Materials; Shaw, R. W., Brill, T. B., Thompson, D. L., Eds.; World Scientific, 2005; pp 303–333.

(44) Trzciński, W. A.; Belaada, A. 1,1-Diamino-2,2-Dinitroethane (DADNE,FOX-7) - Properties and Formulations. 2016, 13 (2), 527–544.

(45) Storm, C. B.; Stine, J. R.; Kramer, J. F. Sensitivity Relationships in Energetic Materials. In Chemistry and Physics of Energetic Materials; Bulusu, S. N., Ed.; Springer, Dordrecht, 1990; pp 605–639.

(46) Kennedy, S. R.; Pulham, C. R. Co-Crystallization of Energetic Materials. In Co-crystals: Preparation, Characterization and Applications; Aakeröy, C. B., Sinha, A. S., Eds.; Royal Society of Chemistry: Cambridge, UK, 2018; pp 231–266.

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Chapter 6

General Conclusions and Future Directions

6.1 General Conclusions

The work in this thesis has explored the development and application of a

model to predict the relative impact sensitivity of a range of EMs. This model

is based on the concept of vibrational up-pumping, which was developed to

rationalise the localisation (and hence intensification) of energy resulting from

mechanical perturbation of a solid. In contrast to previous attempts, the model

in this thesis is based purely on ab initio input. Hence, this work provides a

new approach to predict relative impact sensitivities of EMs.

A model was first constructed for a series of nine azide-based EMs, selected

for investigation on the basis of their diverse structural types and range of

experimental impact sensitivities. Based on literature reports, the relative

sensitivities of these compounds should follow the order: NaN3 ≈ TAGZ

(triaminoguanidinium azide) ≈ NH4N3 < LiN3 < Ba(N3)2< AgN3 < Sn(N3)2, with

the exact position of HN3 and Zn(N3)2 within this order being unknown, except

that they are sensitive to impact. Due to the simplicity of the N3− explosophore,

it was possible to investigate these systems within the framework of a ‘direct’

up-pumping mechanism. Hence, the vibrational normal coordinates of the

explosophore were followed and its electronic structure was monitored. It was

found that the bending modes (δθNNN) led to crossing of the ground-state (𝑆0)

and first triplet-state (𝑇1) potential energy surfaces (PES). Dissociation of the

N-N bond of N3− is favourable for the 𝑇1 PES, and hence δθ𝑁𝑁𝑁 is suggested

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as the target vibrational mode. This was confirmed in the solid state by

monitoring the evolution of the electronic band structure as a function of the

normal coordinates associated with crystalline NaN3. Based on ab initio

phonon dispersion curves, the vibrational up-pumping into the target mode

within each system was considered. In line with previous consideration of the

up-pumping model, the overtone and combination pathway contributions were

isolated. Using only the overtone contributions (which is the method proposed

in previous work1,2), the sensitivity ordering was not well reproduced. While the

prediction does generally place the insensitive compounds at lower sensitivity

than the sensitive compounds, a number of notable exceptions occurred. This

was largely rectified by consideration of the combination contributions (the

method applied in other previous work3–5), although there are again notable

exceptions. It was found that only by considering both mechanisms could the

sensitivity ordering be reproduced.

An up-pumping model was subsequently considered for a series of molecular

energetic materials: 1,1’-azobistetrazole (ABT), hexanitrobenzene (HNB),

1,3,5,7-tetranitro-1,3,5,7-tetrazocane (HMX), 5,5’-hydrazinebistetrazole, 1,1-

Diamino-2,2-dinitroethene (FOX-7), nitrotriazolone (NTO), and

triaminotrinitrobenzene (TATB). However, due to the complexity associated

with dissociation of these molecules, no target vibrational mode could be

identified. Instead, the sensitivity of these materials was explored within the

framework of an ‘indirect’ (or thermal) vibrational up-pumping mechanism. The

calculated vibrational spectra for a subset of the materials (𝛼-FOX-7, NTO,

TATB and 𝛽-HMX) were verified by comparison to inelastic neutron scattering

spectra. A number of models were explored based on ab initio calculation of

the full phonon dispersion curves and are summarised in Table 6.1. The first

two models listed in Table 6.1 can be assessed purely from spectroscopic data.

This may prove useful for rapid screening of newly synthesised materials.

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Table 6.1: Summary of the up-pumping models considered for the treatment of the organic EMs in Chapter 4.

Model Concept Performance Remarks

Vibrational frequency gap

Correlation of the gap in vibrational frequencies between the top of the phonon bath (Ω𝑚𝑎𝑥) and the first doorway mode.

Broad classification of EMs as ‘sensitive’ or ‘insensitive’. Fails to predict sensitivity ordering within each classification. (Figure 4.6)

Doorway mode density Correlate the density of doorway mode states against impact sensitivity

Good agreement with relative sensitivity ordering. Minor mis-ordering. (Figure 4.7)

Overtone Excitation Correlate overtone up-pumping and projection onto doorway modes.

Most successful based on N=2,3 (i.e. two fastest) overtones (Figure 4.10). Excellent agreement with experiment across structurally similar compounds.

Combination Excitation Correlate combination up-pumping of all frequencies < 3Ω𝑚𝑎𝑥 .

Very poor. No notable correlation (Figure 4.13)

Two-layer Model Explicitly consider the two stages of up-pumping: (1) overtone population and projection onto doorway frequencies, and (2) combination up-pumping of PDOS resulting from step (1).

Good correlation across structurally similar compounds (Figure 4.15). Excellent correlation if up-pumping is restricted to 2Ω𝑚𝑎𝑥 → 3Ω𝑚𝑎𝑥 (Figure 4.16)

Two-layer Model + T The two-layer model considering all up-pumping based on thermally-populated vibrational bands.

Excellent correlation across all EMs investigated (Figure 4.20).

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Unlike with the azide-based materials, consideration of the combination

pathways performed very poorly at predicting the impact sensitivity ordering

for these materials. Instead, the sensitivity of these EMs appears to correlate

best with the structure of the doorway region, since:

(1) overtone up-pumping of the doorway region performed well on its own;

(2) The two-layered approach led to significant improvements over the pure

combination-based model.

While the model based on the overtone up-pumping of the doorway

frequencies (the model chosen by Bernstein2 and Coffey1) performed well, the

most successful model was that based on the temperature-dependent

construction of the two-layered model. The difference between the

performance of the temperature-independent and temperature-dependent

two-layer models demonstrates the importance of considering more closely the

rate rather than purely the number of up-pumping pathways. This new two-

layered model is constructed from a mixture of the two independent models

that have previously been proposed in the literature, and therefore represents

the first unified approach to predicting the relative impact sensitivities of EMs.

The largest barrier to the successful application of this model is the simulation

of the full phonon dispersion curves, particularly for large organic EMs. It was

promising to find that (due to low vibrational dispersion) the same trends were

observed when only the zone-centre vibrational frequencies were used for the

input. Hence it was possible to add TATP (a highly sensitive material) to the

model. Applying both the overtone up-pumping model and the temperature-

dependent two-layer model, TATP was successfully identified as being the

most sensitive compound in the test set.

In the final chapter of this thesis, the up-pumping model was tested on two

HMX polymorphs: δ - and β -HMX. The δ -form is well known to be more

sensitive to impact than the β-form. The calculated vibrational structure of δ-

HMX was verified by comparison to inelastic neutron scattering spectroscopy.

Consideration of the overtone up-pumping (third model, Table 6.1) and the

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temperature-dependent, two-layer model both suggested that the δ -form

should be considerably more sensitive than the β -polymorph. This clearly

demonstrated that the up-pumping models are sensitive not only to different

molecules, but also to the crystal structure. The series of temperature-related

FOX-7 polymorphs (α-, β- and γ-FOX-7) were therefore explored. As the

structure becomes increasingly layered (α < β < γ) this series offered an

opportunity to explore why layered materials appear less sensitive than non-

layered materials. BAM fall hammer testing of γ -FOX-7 suggested that it

exhibited the same impact sensitivity as α-FOX-7. However, X-ray powder

diffraction measurements showed that upon impact the γ-form transformed to

the α-form, and hence the impact sensitivity of the former could not be directly

measured. The full phonon-dispersion curves for the three polymorphs of FOX-

7 showed that the maximum frequency of the phonon bath (Ω𝑚𝑎𝑥) decreased

with increased layering. The flat, low frequency Ω𝑚𝑎𝑥 is shared by TATB (the

other insensitive layered material studied here). This appeared to be

responsible for the decrease in predicted impact sensitivity of the FOX-7

polymorphs in the sequence α > β > γ, according to both the overtone up-

pumping and temperature-dependent two-layer models. Thus, the vibrational

up-pumping model offers a new mechanism to rationalise the decreased

sensitivity of layered materials.

With both datasets based on the ‘indirect’ up-pumping mechanism, it is

possible to consider the trends of both Chapters 4 and 5 together. This is done

based on the two most successful models (see Figure 6.1): the overtone up-

pumping model (Model 3 in Table 6.1) and the temperature-dependent two-

layer model. Both models reveal a clear trend between experimental impact

sensitivity and that predicted by the up-pumping contributions. In both cases,

the highly sensitive compounds exhibit considerably larger up-pumping values

than the low sensitivity materials. The δ-form of HMX is predicted to have an

impact sensitivity similar to TATP in both models, and γ-FOX-7 is predicted as

being slightly more sensitive than TATB in both models. Overall, these models

offer a remarkable correlation between experimental impact sensitivity and

predicted sensitivity, across a broad range of EMs and explosophores.

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Figure 6.1: Final predicted sensitivity order for the molecular energetic materials. Note that in all

cases, complete phonon dispersion curves are used, except for TATP and δ-HMX, for which Γ-

point density of states are used. (A) Impact sensitivity based on the overtone up-pumping model

(Model 3 in Table 6.1). (B) Impact sensitivity based on the temperature-dependent two-layer

model (T = 300 K). The difference in y-axis scale results from the number of up-pumping pathways

considered in each case, and the addition of temperature in (B).

However, there do remain some minor discrepancies in the model, particularly

for the most sensitive compounds. Furthermore, some differences also exist

between the overtone and two-layered predictions. The relative ordering of

TATP, δ -HMX and HNB changes between the overtone and two-layered

approach, as shown in Figure 6.1. Noting that TATP is experimentally more

sensitive than HNB6,7 it can be suggested that the two-layered approach is

more successful. The two-layered approach also performs better for the

ordering of β-HMX with respect to the FOX-7 systems. ABT is consistently

predicted to be less sensitive than HNB. However, the nature of the

experimentally reported sensitivity for ABT8 is not known and may therefore

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represent the limiting impact energy. In this case, the ordering of ABT would

not be anomalous.

In summary, the work in this thesis has demonstrated that a vibrational up-

pumping model, based solely on ab initio input, can predict the relative

sensitivity of EMs across a range of materials and explosophores. Within the

direct vibrational up-pumping model (based on a target frequency) used for the

azide materials, a combination of both literature models was required to obtain

a successful prediction of impact sensitivity, hence unifying the literature

discrepancy. Description of the more complex organic EMs required

development of a variety of new models within the framework of an indirect up-

pumping mechanism. Of these models, the newly developed temperature-

dependent two-layered approach appears to be the most successful. This

model comprises aspects of both up-pumping models previously described in

the literature and successfully unifies them into a highly successful approach.

This two layered model proved capable of successfully ordering the impact

sensitivity of a range of materials, as well as of polymorphic systems.

6.2 Future Directions

The individual challenges associated with further development of the up-

pumping model presented in this thesis are presented in each chapter. In these

closing remarks, it is instead worth considering some of the ‘real-world’

applications and new research directions to which this thesis may lead.

There is currently considerable effort being devoted to the development of new

EMs. While the aim is always to develop EMs with enhanced performance, it

is no longer sufficient to consider performance in isolation. New constraints are

now in place, with particular emphasis on the development of insensitive

munitions (IMs). The development of IMs, however, is particularly challenging

as there remains no fundamental insight into what physical or chemical

parameters define sensitivity. Thus, current methods in EM research require

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(often lengthy) synthesis of new EMs and experimental testing of their

sensitivity properties. However, without a priori insight into the physical

properties of a new EM, its synthesis is not only a financial risk, but is

accompanied by potentially serious risks to health and safety.

Modern quantum chemical methods are able to correctly predict molecular and

crystalline structure, vibrational and thermodynamic properties, interaction

energies, amongst a plethora of other properties. Typically, these calculations

run over the period of days to weeks, can be run in parallel, and are

comparatively cheap as an alternative to experiment. Thus, if a method were

known that was capable of predicting sensitivity properties, one could in

principle design an IM in silico, with a full assessment of its sensitivity

properties, without ever needing to set foot in the laboratory until a promising

candidate was found.

The vibrational up-pumping model developed in this thesis is one such model.

It was demonstrated to be capable of predicting the relative sensitivity ordering

of a broad range of EMs based on knowledge of the crystal structure. In its

current form, this model makes it possible to predict the sensitivity of materials

which may be difficult to obtain in large quantities. High-pressure phases are

of particular note. During a detonation, immense pressures are experienced at

the shock front, which are well above the pressures typically required to induce

structural phase transitions in organic EMs.9–11 The reactivity of a material to

mechanical perturbation may therefore change during detonation. Thus,

understanding the relative sensitivity of pressure-related polymorphs may

prove crucial for understanding the detonation properties of polymorphic EMs.

An excellent example of this is RDX, which undergoes numerous high-

pressure phase transitions.11–13 In other cases, new polymorphic forms may

crystallise under pressure,14 and thus it may not be possible to easily prepare

large quantities for testing. The model developed in this thesis can therefore

be applied to predict the sensitivity of a new phase, and make judgement as

to whether purification of this phase should be pursued further. Alternatively,

polymorphic phases may appear as non-isolable impurities within a powder. If

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this impurity acts to sensitise the mixture (e.g. δ-HMX impurities in β-HMX

samples15) additional work must be done to remove it. Hence, the up-pumping

model offers a means to new validation methods for material composition.

Finally, as demonstrated in this thesis, the experimental impact sensitivity

testing of polymorphic materials can prove very challenging, and perhaps

impossible in some cases. The up-pumping model can therefore be used in

parallel with experimental results to assist in their interpretation.

A particularly promising application of the model developed in this thesis is in

parallel with crystal structure prediction. It is now possible to predict the

potential crystal structures of complex organic molecules to a good degree of

accuracy.16 The combination of crystal structure prediction with sensitivity

prediction truly offers the way to a new paradigm in EM research. New

molecules could be completely designed in silico, their crystal structures

predicted, and the up-pumping model applied to generate a list of sensitivities.

In doing so, the work in this thesis would open the door to a complete

restructuring of the way EM research is performed. As further developments

are made on the up-pumping model (e.g. inclusion of electronic effects,

vibrational response to pressure, defects, etc.), its performance will surely

improve. As it does, the reality of making in silico EM design a reality becomes

closer.

6.3 References

(1) McNesby, K. L.; Coffey, C. S. Spectroscopic Determination of Impact Sensitivities of Explosives. J. Phys. Chem. B 1997, 101 (16), 3097–3104.

(2) Bernstein, J. Ab Initio Study of Energy Transfer Rates and Impact Sensitivities of Crystalline Explosives. J. Chem. Phys. 2018, 148 (8), 084502.

(3) Ye; Koshi, M. Theoretical Studies of Energy Transfer Rates of Secondary Explosives. J. Phys. Chem. B 2006, 110 (37), 18515–18520.

(4) Ye, S.; Tonokura, K.; Koshi, M. Energy Transfer Rates and Impact Sensitivities of Crystalline Explosives. Combust. Flame 2003, 132 (1–2), 240–246.

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(5) Fried, L. E.; Ruggiero, a J. Energy-Transfer Rates in Primary, Secondary, and Insensitive Explosives. J. Phys. Chem. 1994, 98 (39), 9786–9791.

(6) Storm, C. B.; Stine, J. R.; Kramer, J. F. Sensitivity Relationships in Energetic Materials. In Chemistry and Physics of Energetic Materials; Bulusu, S. N., Ed.; Springer, Dordrecht, 1990; pp 605–639.

(7) Gamage, N. D. H.; Stiasny, B.; Stierstorfer, J.; Martin, P. D.; Klapötke, T. M.; Winter, C. H. Less Sensitive Oxygen-Rich Organic Peroxides Containing Geminal Hydroperoxy Groups. Chem. Commun. 2015, 51 (68), 13298–13300.

(8) Klapötke, T. M.; Piercey, D. G. 1,1′-Azobis(Tetrazole): A Highly Energetic Nitrogen-Rich Compound with a N10 Chain. Inorg. Chem. 2011, 50 (7), 2732–2734.

(9) Klapötke, T. M. Chemistry of High-Energy Materials, 2nd ed.; Klapötke, T. M., Ed.; De Gruyter: Berlin, 2012.

(10) Millar, D. I. A.; Maynard-Casely, H. .; Kleppe, A. .; Marshall, W. G.; Pulham, C. .; Cumming, A. . Putting the Squeeze on Energetic Materials - Structural Characterisation of High-Pressure Phase of CL-20. CrystEngComm 2010, 12, 2524–2527.

(11) Millar, D. I. A.; Oswald, I. D. H.; Barry, C.; Francis, D. J.; Marshall, W. G.; Pulham, C. R.; Cumming, A. S. Pressure-Cooking of Explosives—the Crystal Structure of ε-RDX as Determined by X-Ray and Neutron Diffraction. Chem. Commun. 2010, 46 (31), 5662–5664.

(12) Davidson, A. J.; Oswald, I. D. H.; Francis, D. J.; Lennie, A. R.; Marshall, W. G.; Millar, D. I. A.; Pulham, C. R.; Warren, J. E.; Cumming, A. S. Explosives under Pressure—the Crystal Structure of γ-RDX as Determined by High-Pressure X-Ray and Neutron Diffraction. CrystEngComm 2008, 10 (2), 162–165.

(13) Millar, D. I. A.; Oswald, I. D. H.; Francis, D. J.; Marshall, W. G.; Pulham, C. R.; Cumming, A. S. The Crystal Structure of Beta-RDX-an Elusive Form of an Explosive Revealed. Chem. Commun. 2009, No. 5, 562–564.

(14) Fabbiani, F. P. A.; Pulham, C. R. High-Pressure Studies of Pharmaceutical Compounds and Energetic Materials. Chem. Soc. Rev. 2006, 35 (10), 932–942.

(15) Achuthan, C. P.; Jose, C. . Studies on HMX Polymorphism. Propellants, Explos. Pyrotech. 1990, 275, 271–275.

(16) Price, S. L. Predicting Crystal Structures of Organic Compounds. Chem. Soc. Rev. 2014, 43 (7), 2098–2111.

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Appendix A

Publications

In This Thesis

Published

1. Michalchuk, A.A.L., Fincham, P.T., Portius, P., Pulham, C.R., and Morrison, C.A., A Pathway

to the Athermal Impact Initiation of Energetic Azides, J Phys. Chem. C. 2018, 122(34), 19395-

19408

2. Michalchuk, A.A.L. Rudic, S., Pulham, C.R., and Morrison, C.A., Vibrationally Induced

Metallisation of the Energetic Azide α-NaN3, Phys Chem Chem Phys, 2018 , 20, 29061-29069

In Preparation

3. Michalchuk, A.A.L., Piggott, S., Rudic, S., Pulham, C.R., and Morrison, C.A. Impact

Sensitivity of Polymorphic Energetic Materials: The Curious Case of FOX-7. In preparation

4. Michalchuk, A.A.L., Triestman, M., Rudic, S., Pulham, C.R. and Morrison, C.A, Vibrational

up-pumping in molecular energetic materials. In preparation

5. Michalchuk, A.A.L., Rudic, S., Pulham, C.R. and Morrison, C.A. Predicting Impact Sensitivity

from Inelastic Neutron Scattering Spectroscopy. In preparation

Not In This Thesis

Published

1. Michalchuk, A.A.L., Tumanov, I.A., Boldyreva, E.V., Complexities of Mechanochemistry,

Elucidation of Processes Occurring in Mechanical Activators via Implementation of a Simple

Organic System, CrystEngComm, 2013 15, 6403. Hot Article

2. Michalchuk, A.A.L., Tumanov, I.A., Drebushchak, V.A. and Boldyreva, E.V., Advances in

elucidating mechanochemical complexities via implementation of a simple organic system,

Faraday Disucss, 2014, 170, 311

3. Bell, N.G.A., Michalchuk, A.A.L., Blackburn, J.W., Graham, M.C., and Uhrin, D., Isotope-

filtered 4D NMR spectroscopy for structure determination of humic substances, Angew. Chem.

Int. Ed., 2015, 54, 8382

4. Tumanov, I.A., Michalchuk, A.A.L., Politov, A.A, Boldyreva, E.V., and Boldyrev V.V.,

Inadvertent liquid assisted grinding: a key to ‘dry’ organic mechano-co-crystallisation?

CrystEngComm, 2017 19, 2830

5. Tumanov, I.A., Michalchuk, A.A.L., Politov, A.A., Boldyreva, E.V., and Boldyrev, V.V,

Inhibition of organic mechanochemical synthesis by water vapor, Doklady Chem., 2017 472.

6. Michalchuk, A.A.L., Tumanov, I.A., Konar, S. Kimber, S.A.J., Pulham, C.R. and Boldyreva,

E.V. Challenges of Mechanochemistry: Is In Situ Real‐Time Quantitative Phase Analysis

Always Reliable? A Case Study of Organic Salt Formation, Adv. Sci., 2017, 4, 1700132

250

7. Bouvart, N., Palix, R.-M., Arkhipov, S.A., Tumanov, I.A., Michalchuk, A.A.L., and Boldyreva,

E.V. Polymorphism of chlorpropamide on liquid assisted mechanical treatment: choice of

liquid and type of mechanical treatment matter, CrystEngComm, 2018, 20, 1797

8. Michalchuk, A.A.L., Hope, K.S., Kennedy, S.R., Blanco, M.V., Boldyreva E.V. and Pulham,

C.R., Ball-free mechanochemistry: In situ real time monitoring of pharmaceutical co-crystal

formation by resonant acoustic mixing, Chem. Commun. 2018, 54, 4033.

9. Michalchuk, A.A.L., Tumanov, I.A., and Boldyreva, E.V. The effect of ball mass on the

mechanochemical transformation of a single-component organic system: anhydrous caffeine,

J. Mat. Sci, 2018, 53 (19), 13380-13389.

10. Zakharov, B.A., Michalchuk, A.A.L., Morrison, C.A. and Boldyreva, E.V. Anisotropic Lattice

softening near the structural phase transition in the thermosalient crystal 1,2,4,5-

tetrabromobenzene, Phys Chem Chem Phys, 2018, 20, 8523 Hot Article

11. Tantardini, C. and Michalchuk, A.A.L. Dess-martin periodinane: The reactivity of a λ5-iodane

catalyst explained by topological analysis Int. J. Quantum Chem. 2019, Accepted.

In Preparation

1. Konar, S, Michalchuk, A.A.L., Sen, N., Bull, C., Morrison, C.A., Pulham, C.R. High pressure

neutron diffraction and DFT-D study of TNT polymorphs. In preparation

251

Appendix B

Conferences and Courses

Year 1

This year was dedicated to training under the EPSRC Doctoral Training Centre

in Innovative Manufacturing in Continuous Manufacturing and Crystallisation.

Training included 11 training weeks from Oct 2014 – May 2015 distributed

across the centre partner institutions.

Conferences

1. MechChem, July 2015. Montpelier, France. Poster Presentation:

‘To beat or not to beat. The role of impact frequency on

mechanochemical transformations’

Workshops

1. Resodyn Technical Exchange. Aug 2015. Butte, Montana, USA.

Courses (University of Edinburgh)

1. Electronic Structure Theory and Classical Simulation Methods

2. Computational Modelling of Materials

3. Computer-Aided Drug Design

Year 2

Conferences

1. Hot Topics in Solid State Chemistry: All Russian (with international

participants) Conference. Novosibirsk, Russian Federation. October

2015. Oral Presentation: ‘In situ real time monitoring of

mechanochemical transformations’

2. International Research Conference on Expanding Frontiers of RNA

Chemistry and Biology, Novosibirsk, Russian Federation. Nov 2016.

Oral Presentation: ‘Mechanochemical methods for organic systems’

252

3. 31st European Crystallographic Conference. Basel, Switzerland. Aug.

2016. Oral Presentation (on behalf of E Boldyreva): ‘Crystallography

in Education’

4. 31st European Crystallographic Conference. Basel, Switzerland. Aug.

2016. Poster Presentation: ‘In situ real-time monitoring of

mechanochemical salt formation’

Workshops

1. ISIS Neutron Training Course, 12-21 Apr 2016

Year 3

Conferences

1. Crystal Forms, University of Bologna, Italy. 4-6 June 2017

2. 24th IUCr, Aug 2017. Hyderabad, India. Poster Presentation: ‘The Big

Bang Theory. Towards Predicting Impact Sensitivity of Energetic

Materials’

Year 4

Conferences

1. 49th Conference of the Fraunhofer ICT. June 2018. Oral

Presentation: ‘The Big Bang Theory. Towards Predicting Impact

Sensitivity of Energetic Materials’